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Temperature and dislocation stress field models of the LEC growth of gallium arsenide Schvezov, Carlos Enrique 1986-08-18

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TEMPERATURE AND DISLOCATION STRESS FIELD MODELS OF THE LEC GROWTH OF GALLIUM ARSENIDE by CARLOS ENRIQUE SCHVEZOV Lie. en Fisica, Universidad Nacional de Rosario, Argentina, 1975 M.A.Sc. in Metallurgy, University of British Columbia, Canada, 1983 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Metallurgical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1986 © Carlos Enrique Schvezov, 1986 S In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) 11 ABSTRACT The temperature fields and resulting stress fields have been calculated for a growing GaAs crystal produced by the LEC process. The calculations are based in a finite element numerical thermoelastic stress analysis. The calculated temperature fields have been compared to reported experimental measurements with good agreement. The stress fields have been used to calculate the resolved shear stresses, in the growing crystal, from which the dislocation density and distribution were determined. Using the model the effects of a range of growth and environmental parameters on the dislocation density and distribution were determined. Theses parameters include crystal length, crystal diameter, cone taper, boron oxide thickness, gas pressure, solid/liquid interface shape, vertical temperature gradients and others. The results show that the temperature distribution in the gas surrounding the crystal, and the boron oxide thickness, were critical factors in determining the dislocation density and distribution in the crystal. The crystal radius, crystal length and interface curvature also strongly influenced the dislocation configuration. After crystal growth, the dislocation density at the end of the crystal was strongly influenced by the cooling procedure adopted. The dislocation distribution on cross-sections of the crystal exhibited two-fold, four-fold and eight-fold symmetry depending on growth and cooling conditions and position in the crystal. Ill TABLE OF CONTENTS Page ABSTRACT II TABLE OF CONTENTS Ill LIST OF FIGURES XLIST OF TABLES XXV LIST OF SYMBOLS XXVI ACKNOWLEDGEMENTS XXX CHAPTER 1 INTRODUCTIONCHAPTER 2 THE GROWTH OF BULK GaAs CRYSTALS 4 2.1 The Liquid Encapsulated Czochralski Technque ( LEC ) 4 2.2 Classification of Defects in LEC grown GaAs crystal 7 2.3 Dislocations in LEC-GaAs 8 2.4 Effect of Dislocations on Properties of GaAs Devices 11 CHAPTER 3 THE ORIGIN OF DISLOCATIONS IN LEC GaAs CRYSTALS 12 CHAPTER 4 DISLOCATION DENSITY AND CRYSTAL GROWTH 17 IV 4.1 Stresses in the Crystal Due to Thermal Gradients u 17 4.1.1 Crystal Dimensions. 18 4.1.2 Thermal Conditions During Crystal Growth 20 4.2 Alloy Addition to the GaAs 22 CHAPTER 5 MODELS OF DISLOCATION GENERATION IN GaAs 29 5 . 1 Stress Fields . 25.2 Calculation of Dislocation Densities 36 CHAPTER 6 OBJECTIVES 40 CHAPTER 7 FORMULATION OF THE MODELS 43 7.1 Model of crystal growth 45 7.1.1 Temperature Field7.1.1.1 Governing Equations 45 7.1.1.2 Finite Element Equations 8 7.1.2 Stress Field Model 53 7.1.2.1 Governing and Element Equations 57.1.2.2 Requirements for the interpolation functions.... 57 7.1.2.3 Formulas for Stress Calculations 59 7.1.2.4 Linear Elements 60 7.1.2.5 Quadratic Element 1 7.1.3 Von Mises and Resolved Shear Stresses 63 7.2 Modelling for Cooling After growth 7 V 7.3 Analytical Solutions 71 7.3.1 Analytical QSS Temperature Field 77.3.2 Analytical solutions for the Stress Field 73 7.3.2.1 Plane Strain Approximation 77.3.2.2 Axisymmetric Solutions 4 CHAPTER 8 EVALUATION OF THE MODELS ' 76 8.1 Programming and Input Parameters 78.1.1 Input Parameters 86 8.1.2 Numerical Evaluation of the Analytical Solutions 93 8.2 Comparison of Model Predictions with Analytical Solutions8.2.1 Temperature Field 93 8.2.2 Stress Fields 7 8.3 Comparison of Model Predictions with Temperature Measurements 108.4 The Temperature Model for Cooling - Programming and Convergency 115 CHAPTER 9 RESULTS AND ANALYSIS 120 9.1 Cone Angle 1 9.1.1 Effect of Cone Angle on Stress Symmetry 133 9.1.2 Effect of Thermal Conditions 146 9.1.3 Effect of the Heat Transfer Coefficient 155 9.1.4 Effect of Non-linearity in the Temperature Profile 161 9.2 Crystal Length 170 VI 9.2.1 Effect of crystal Length on Stress Symmetry 175 9.3 Crystal Radius 190 9.3.1 Effect of Radius on Stress Symmetry 197 9.4 Growth Velocity 199 9.5 Thermal Conditions 202 9.5.1 Radius 27.5 mm 5 9.5.1.1 Boron oxide thickness 21.0 mm 209.5.1.2 Boron oxide thickness 40.0 mm 207 9.5.2 Radius 40.0 mm 219.5.2.1 Boron oxide thickness 21 mm 217 9.5.2.2 Boron oxide thickness 40 mm 219.5.2.3 Boron Oxide thickness 50 mm 221 9.6 Gas Pressure and Composition 227 9.7 Curvature of the Solid-Liquid Interface 235 9.7.1 Convex interface 236 9.7.1.1 Crystal Length9.7.1.2 Effect of Encapsulant Thickness and Gradient 245 9.7.1.3 Effect of crystal Radius 251 9.7.2 Concave Interface 253 CHAPTER 10 RESULTS FOR COOLING AFTER GROWTH 257 10.1 Ambient Temperature 1000°C 258 10.1.1. Argon 2510.1.1.1 Initial Gradient GC = 70°C/cm 258 10.1.1.2 Initial Gradient GC/2 = 35°C/cm 269 10.1.1.3 Initial Gradient GC/4 = 17.5°C/cm 276 VII 10.1.2. Boron Oxide 278 10.2 Ambient Temperature 800°C/cm 283 10.3 Analysis of the Results for Cooling 283 CHAPTER 11 SUMMARY AND CONCLUSIONS 290 REFERENCES 306 APPENDIX I EFFECT OF DISLOCATIONS ON GaAs AND DEVICES 320 1.1 Effect of Dislocation on Properties 321.2 Dislocations and Fermi Level 327 APPENDIX II Integration by Parts of the Element Equation for the Temperature Field 330 APPENDIX III Evaluation of the Matrix Elements for the Temperature Field 332 APPENDIX IV Evaluation of Stiffness Matrix Elements for Linear Elements " 338 APPENDIX V Quadratic Element Calculations 340 VIII APPENDIX VI Calculation of Resolved Shear Stresses 353 APPENDIX VII Temperature Field During Cooling 362 APPENDIX VIII Analytical Solution of the Temperature Field 367 APPENDIX IX Plane Strain Analytical Stresses 372 APPENDIX X ANALYTICAL AS ISYMMETRIC SOLUTIONS FOR STRESSES 375 X.l Axisymmetric Temperature Field 372 X. 2 Analytical Axisymmetric Solution for 6 = - p ... 385 APPENDIX XI COMPUTER PROGRAMS 390 XI. 1 Mesh Generator 39XI.2 Finite Linear Element Program for the Temperature Calculations during Growth 394 XI.3 Finite Linear Element Program for the Stress Calculations 402 XI.4 Finite Quadratic Element Program for the Stress Calculations 41XI.5 Program for the Calculations of the MRSS-CRSS in (010) Sections 425 XI.6 Program for the Calculations of MRSS-CRSS in (001) Sections 428 IX XI. 7 Program for the Numerical Evaluation of Temperatures During Cooling 432 APPENDIX XIII XII. 1 Program for the Numerical Evaluation of Analytical Axisymmetric Temperature Fields 437 XII.2 Program for the Numerical Evaluation of Analytical Analytical Axisymmetric Stresses for Radial Temperature Fields 439 XII. 3 Program for the Numerical Evaluation of Analytical Axisymmetric Stresses for Axisymmetric Temperature Fields 441 APPENDIX XIII TEMPERATURE AND STRESS PLOTS FROM THE COMPUTER PROGRAMS OUTPUT (GROWTH) 444 XI11 . 1 Cone Angle CL, 10 mm ; R, 20 mm : B, 10 mm ; AP, 30 atm. ; BG, 100 C/cm ; AG, 50 C/cm 444 XIII. 2 Crystal Length R, 27.5 mm ; B.21 mm ; AP, 30 atm. ; CA, 30° ... 521 XIII.3 Crystal Radius R, 40 mm ; B, 21 mm ; CA, 30° ; AP, 30 atm 575 XIII.4 Growth Velocity R 27.5 mm ; CL , 55 mm ; CA, 30° ; B, 21 mm 617 XIII.5 Thermal Condition R, 27.5 mm CL, 55 mm ; CA, 30 ; AP, 30 atm 624 XIII.6 Thermal Conditions R, 40 mm CL, 80 mm ; CA, 30° ; AP , 30 atm 655 XIII.7 Argon Pressure, 2 atmospheres. R, 27.5 mm ; CL, 55 mm ; CA, 30° ; B, 21 mm .... 690 XIII.8 Curvature of the Solid/Liquid Interface CA. 30° ; AP, 30 atm 703 X APPENDIX XIV TEMPERATURE AND STRESS PLOTS FROM THE COMPUTER PROGRAMS OUTPUT (COOLING) 761 XIV.1 Ambient Temperature 800°C, Argon 76XIV.2 Ambient Temperature 1000°C 799 XI LIST OF FIGURES Fig. 2.1 Schematic of a LEC pulling chamber 5 Fig. 3.1 Transmission X-ray topography in GaAs - (a) and (b) undopgd ; (c) Te-doped, (110) axial section 13 Fig. 4.1 Radial dislocation density profiles across wafers obtained from the front, middle, and tail of a crystal. The radial profiles are "W" shaped, and the average EPD increases from the front to the tail 19 Fig. 4.2 Mean density of "grown-in" dislocations in GaAs single crystals (20-25 mm in diameter) grown by the Czochralski LEC Technique, as a function of dopant concentration : (1) Te, (2) Sn, (3) In, (4) Zn 23 Fig. 4.3 Hypothetical distribution of the dopant (In) along the pulling axis of the ingot. The indium concentration oscillates around a mean value which is proportional to the distance from the top of the ingot 26 Fig. 5.1 (a) and (b) - Isotherms and shear stress topography in two gallium arsenide single crystal grown under different conditions (c) - Distribution of Shear Stresses T ^ T , and (d) - dislocations density over the cross-sections over galium arsenide single crystals... 30 Fig. 5.2 T R S S contours for the top wafer of a <001> GaAs boule 33 Fig. 5.3 Distribution of the calculated dislocation density in a gallium arsenide single crystal being grown in the < 111 > direction for slip systems with Burgers dislocation vectors perpendicular 34 XII Fig. 7.1 Flow chart of the model used to derive the stress fields 44 Fig. 7.2 Crystal configuration and coordinate system used in the mathematical model 47 Fig. 8.1 Flow chart of the computer program to calculate the temperature field in the crystal using a finite element method 78 Fig. 8.2 Flow chart of the computer program for the stress calculations using finite linear and quadratic elements 81 Fig. 8.3 Flow chart of the mesh generator computer program 84 Fig. 8.4 Flow chart of the computer program to calculate and plot RSS in a vertical (010) plane 87 Fig. 8.5 Flow chart of the computer program to calculate and plot RSS in a (001) plane 88 Fig. 8.6 Estimated radiative and convection heat transfer coefficients for GaAs/B20 (1), He (g), N2 (g) and A (g) as a function of ambient temperature. The numerical labels are the product of the carrier concentration X crystal diameter in units of cm 90 Fig. 8.7 Total heat transfer coefficient h, in B2®3 an(* argon as a function of the ambient temperature T (1) Total heat transfer coefficient in ^2°3' ( § ) Total heat transfer coefficient in argon pressurized at 30 atm 90 Fig. 8.8 Temperature dependence of the critical stress for dislocation generation in GaAs : (1) Te-doped material, n = 2 X 10*® cm (2) Te-doped material, n = 7 X 10 cm XIII (3) Zn-doped material, (4) undoped material.. 9 X 1018 cm 3 92 Fig. 8.9 A comparison of finite element and analytical calculated temperature curves for h 0.3 cm 94 Fig. 8.10 A comparison of finite element and analytical calculated temperature curves for h 0.6 cm" 95 Fig. 8.11 Mesh employed in the calculations of the temperature fields shown in Fig. 8.9 and 8.10 Number of nodes = 45. Number of elements = 64 96 Fig. 8.12 Calculated radial stresses as a function of r/r for radial temperature fields (1) Finite element with averaged element (2) Finite element with nodal temperatures (3) Analytical plane strain (4) Ana 1ytica1-axisymmetric 98 Fig. 8.13 Calculated azimuthal stresses as a function of r/r for radial temperature fields (1) Finite element with averaged element temperatures (2) Finite element with nodal temperatures (3) Ana 1ytica1-p1ane strain (4) Analytical-axisymmetric 99 Fig. 8.14 Calculated axial stresses as a function of r/r for radial temperature fields (1) Finite element with averaged element temperatures (2) Finite element with nodal temperatures (3) Ana 1ytical-piane strain (4) Analytical-axisymmetric 100 Fig. 8.15 Four steps in the mesh refinement used to analyse the convergency of the finite element stress calculations. (a) NN = 15, NE = 16 (b) NN = 45, NE = 16 (c) NN = 45, NE = 64. (d) NN = 153, NE = 256 (b) Quadratic elements (a), (c) and (d) linear elements 102 Fig. 8.16 Calculated radial stresses as a function of r/r for different numbers of nodes and sizes of element (1) Finite element with averaged XIV element temperature (2) Finite element with nodal temperatures. L = Linear element, Q = Quadratic element 103 Fig. 8.17 Calculated radial stresses as a function of r/r for axisymmetric thermal fields (l)°Finite element with averaged element temperatures (2) Finite element with nodal temperatures (3) Ana 1ytica1-p1ane strain (4) Analytical-axisymmetric 105 Fig. 8.18 Position of thermocouples in GaAs crystal. B = Boric oxide layer, argon pressure 3.04 MPa. Reference 218 108 Fig. 8.19 Temperatures measured with thermocouples 2, 3 and 4 in Figure 8.18 as a function of the relative position of the thermocouples with the interface. Reference 218 110 Fig. 8.20 Measured and calculated temperatures along the crystal axis at four crystal lengths Ill Fig. 8.21 Measured and calculated temperatures adjacent to the outside surface of the crystal at four crystal lengths. The measured ambient temperature is also shown 112 Fig. 8.22 Measured and calculated axial temperature gradients along the crystal axis at four crystal 1 engths 113 Fig. 8.23 Flow chart of the computer program for the numerical evaluation of the analytical temperature fields during cooling of the crystal 117 Fig. 8.24 (a) Typical temperatugeQfie1d obtained during cooling units are 10 C. (b) and (c) Von Mises stress contours (MPa) for the temperature field given in (a). (b) NN = 451, NE = 800. (c) NN = 1105, NE = 2048 118 Von Mises Stress contours (MPa) in vertical planes for five cone angles, (a) 7.1°, (b) 30°, (c) 45°, (d) 54. 7°, (e) 65°. Cone surface in (d) coincides with a (111) plane. Crystal radius, 20 mm ; crystal length, 10 mm ; B2^3 thickness, 10 ram ; B 0^ gradient, 100 C/cm ; argon pressure, 30 atm. ; argon gradient, 50°C/cm Maximum resolved shear stress (MRSS) contours in MPa, in vertical (010) planes for five cone angles. (a) 7.1°, (b) 30°, (c) 45°, (d) 54.7°, (e) 65 . Cone surface in (d) coincides with a (111) plane. Conditions are the same as in Figure 9.1 Total resolved shear stress (TRSS) contours in MPa for the 45 cone angle crystal. Compare the large stress levels of the TRSS with the stress levels for same crystal shown in Figure 9.1(c) for the VMS and Figure 9.2(c) for the MRSS Contours of the MRSS (MPa) in excess of (a) CRSS (yield) ; (b) CRSS (MB) ; (c) CRSS (MBTe). Shaded regions indicate areas in which the MRSS is less than the CRSS Slip mode distribution in the (010) plane for the 45 cone angle crystal corresponding to the MRSS distribution shown in Figure 9.2(c) Contours in MPa of the MRSS in excess of (a) CRSS (Yield) ; (b) CRSS (MB) and (c) CRSS (MBTe). In most of the crystal the MRSS is larger than the critical values. The bump shaped contours (A) give complex stress distributions in perpendicular cross-sections MRSS contours (MPa) in cross-sections perpen dicular to the crystal axis for five cone angles, (a) 7.1°, (b) 30°, (c) 45°, (d) 54.7°, (e) 65°. Section (a-d) are 7.5 mm from the interface and section (e) is 8.0 mm from the interface. The horizontal direction corresponds to the [100] direction and the vertical direction corres ponds to the [010] direction XVI Fig. 9.8 Contours of the MRSS-CRSS (Yield) (MPa) for the 65 cone angle crystal. MRSS contours are shown in Figure 9.7(e). At high cone angles MRSS levels are larger than CRSS at the centre and outside part of the wafer 142 Fig. 9.9 Stress contours (MPa) in cross-section 2.5 mm from the interface, (a) and (b) in a 7.1° cone angle crystal, (c) in a 45° cone angle crystal, (a) MRSS contours, (b) and (c) MRSS-CRSS (Yield). For the 7.1 cone angle there is eight-fold symmetry at the centre in (a) which is not seen in (b) because stress levels are less than CRSS. For the 45 cone angle there is four-fold symmetry 145 Fig. 9.10 Slip mode distribution in a (001) plane corres ponding to the MRSS distribution shown in Figure 9.7(c) for the 45 cone angle crystal at 7.5 mm from the interface. The eight-fold distribution of the mode at the edge of the section is associated with the eight-fold symmetry of the MRSS 147 Fig. 9.11 Temperature distribution as a function of external temperature gradient for a 45° cone angle crystal. Thermal gradients in the boron oxide are : (a) 50 C/cm, (b) 100 C/cm, (c) 200 C/cm angQ(d) 400 C/cm. Temperatures are given in 10 C. For low gradients isotherms are nearly flat and slightly convex. For larger gradients isotherms are curved and concave 149 Fig. 9.12 Stress contours (MPa) in a (010) plane for a 45° cone angle crystal grown with a 50°C/cm gradient in the boron oxide, (a) MRSS contours does not show the bump shaped stress distribution below the shoulder in (a). The MRSS-CRSS (Yield) is positive only in a few regions in (b) 151 Fig. 9.13 (a) MRSS contours and (b) MRSS-CRSS (Yield) contours for a 200 C/cm gradient in the boron oxide. Units are in MPa. MRSS stress are large below the shoulder in (a). The MRSS-CRSS is greater than zero in most of the crystal in (b). 152 Fig. 9.14 (a) MRSS contours, (b) MRSS-CRSS (Yield) contours, (c) Slip mode distribution. Gradient XVII in the boron oxide is 400°C/cm. In (a) contours are similar to those obtained for a 200°C/cm gradient. Stress levels have doubled. In (b) only a small area in the seed developed stresses less than the CRSS. In (c) the mode distribution is similar to that shown in Figure 9.5 for a 100°C/cm gradient 154 Fig. 9.15 MRSS contours in MPa in a cross-section at 7.5 mm from the interface for the crystal shown in Figure 9.14(a). The symmetry is similar to that shown in Figure 9.7(c) for a 100°C/cm gradient 156 Fig. 9.16 Temperature profiles at the surface of 45° cone angle crystals for three diferent conditions. Curve H corresponds to the temperatures calcu lated using the original heat transfer coef ficient values. In curves H / 1.3 and H x 1.3 the original heat transfer coefficient values were divided and multiplied by 1.3 respectively 158 Fig. 9.17 Stress contours in MPa derived from the temperature field obtained using heat transfer coefficient values 1.3 larger than original values. (a) MRSS contours. (b) MRSS-CRSS (Yield) 159 Fig. 9.18 Stress contours in MPa derived from, the temperature field obtained using heat transfer coefficient values 1.3 times smaller than original values. (a) MRSS contours. (b) MRSS-CRSS (Yield). Variations of only 20 * are observed in the maximum MRSS values between (a) and Figure 9.17(a) 160 Fig. 9.19 MRSS (MPa) contours for a crystal growing under four different conditions given in Table 9.3. (a) Run #1 ; (b) Run #2 ; (c) Run #3 ; (d) Run #4. From (a) to (b) the AMRSS doubles. In (d) the bump shape below the shoulder does not appear 165 Fig. 9.20 MRSS (MPa) contours in a (001) plane at a distance of 5.0 mm from the interface in the crystal shown in Figure 9.19(d) (run #4) XVI I I The four-fold symmetry with a slight two-fold symmetry is observed 167 Fig. 9.21 MRSS-CRSS (Yield) contours (MPa) for the four growth conditions shown in Table 9.3 and MRSS contours shown in Figure 9.19 (a) Run #1 ; (b) Run #2 ; (c) Run #3 ; (d) Run #4 169 Fig. 9.22 MRSS (MPa) contours for five crystal lengths. (a) 13.75 mm ; (b) 27.5 mm ; (c) 55.0 mm ; (d) 82.5 mm ; (e) 110.0 mm . Crystal radius is 27.5 mm 172 Fig. 9.23 MRSS-CRSS (yield) (MPa) for five crystal lengths. (a) 13.75 mm ; (b) 27.5 mm ; (c) 55.0 mm ; (d) 82.5 mm ; (e) 110.0 mm . Crystal radius is 27.5 mm 174 Fig. 9.24 Slip mode of the MRSS for the five crystal lengths shown in Figure 9.22 (a) 13.75 mm ; (b) 27.5 mm ; (c) 55.0 mm ; (d) 82.5 mm ; (e) 110.0 mm . Crystal radius is 27.5 mm 176 Fig. 9.25 MRSS-CRSS (Yield) (MPa) in a (001) plane at a distance of 8.25 mm from the cone at four crystal lengths, (a) 13.75 mm ; (b) 27.5 mm ; (c) 55.0 mm ; (d) 82.5 mm. Crystal radius 27.5 mm 182 Fig. 9.26 MRSS-CRSS (Yield) (MPa) in a (001) plane at a distance of 49.5 mm from the cone at two crystal lengths, (a) 55.0 mm ; (b) 82.5 mm. Crystal radius 27.5 mm 185 Fig. 9.27 MRSS-CRSS (Yield) (MPa) in a (001) plane at a distance of 77.0 mm from the cone at two crystal lengths, (a) 82.5 mm ; (b) 110.0 mm. Crystal radius 27.5 mm 188 Fig. 9.28 MRSS (MPa) contours for four crystal lengths. (a) 20.0 mm ; (b) 40.0 mm ; (c) 80.0 mm ; (d) 100.00 mm. Crystal Radius 40.0 mm 193 Fig. 9.29 (a) Axial along the crystal axis and (b) radial thermal gradients as a function of distance XIX from the interface for (1) for a radius of 27 (3) for a radius of 40 (1) 55.0 mm ; (2 ) 40.0 two crystal radius. Curve 5 mm ; and curves (2) and 0 mm. Crystal lengths are mm and (3) 80.0 mm 194 Fig. 9.30 MRSS-CRSS (Yield) (MPa) for four crystal lengths. (a) 20.0 mm ; (b) 40.0 mm ; (c) 80.0 mm ; (d) 100.00 mm. Crystal Radius 40.0 mm 196 Fig. 9.31 Temperature fields for three growth velocities (a) 0.0001 cm/s, (b) 0.001 cm/s and (c) 0.01 cm/s. Temperature given in 10 °C. Little change is observed from (a) to (b). From (b) to (c) gradients have increased. Radius 27.5 mm, length 55 mm, encapsulant thickness 21 mm 201 Fig. 9.32 MRSS-CRSS (Yield) (MPa) for the three tempera ture fields shown in Figure 9.31(a-c) corres ponding to three growth velocities, (a) 0.0001 cm/s, (b) 0.001 cm/s and (c) 0.01 cm/s. In (a) and (b) the stress fields are similar. In (c) the stress distribution and stress values change 204 Fig. 9.33 Temperature profiles along the crystal surface in the environment surrounding the crystal employed in the calculations. The profiles are derived from Figure 8.19, curve 4, reproduced as curve G. The values of gradient are averaged in the crystal length considered. Encapsulant thickness 21 mm 206 Fig. 9.34 MRSS-CRSS (Yield) (MPa) for four temperature profiles with average gradients (a) 33°c/cm, (b) 17°C/cm, (c) ll°C/cm and (d) 8°C/cm. Radius 27.5 mm, length 55 mm. Encapsulant thickness 21 mm 209 Fig. 9.35 Temperature profiles used in the calculations for two boron oxide thicknesses. (a) 40 mm, (b) 50 mm. The profiles are derived from curve G. The values of gradients given are averaged along the crystal length 211 Fig. 9.36 MRSS-CRSS (Yield) (MPa) for three average gradients. (a) 58 C/cm, (b) 29°C/cm, (c) XX 19 C/cm. Radius 27.5 mm, length 55 mm, boron oxide thickness 40 mm 213 Fig. 9.37 MRSS-CRSS (Yield) (MPa) for three average gradients, (a) 50°c/cm, (b) 25°C/cm, (c) 17"C/ cm. Radius 27.5 mm, length 55 mm, boron oxide thickness 50 mm 215 Fig. 9.38 MRSS-CRSS (Yield) (MPa) for two average gradi ents, (a) 14°C/cm, (b) 7 C/cm. Radius 40 mm, length 80 mm, boron oxide thickness 21 mm 218 Fig. 9.39 MRSS-CRSS (Yield) (MPa) for four average gradients, (a) 46 C/cm, (b) 26°C/cm, (c) 15°C/cm, (d) 9°C/cm. Radius 40 mm, length 80 mm, boron oxide thickness 40 mm 220 Fig. 9.40 MRSS-CRSS (Yield) (MPa) for a crystal length of 52 mm comparable to the boron oxide thickness of 50 mm. Radius 40 mm. Average gradient 50 C/cm... 222 Fig. 9.41 MRSS-CRSS (Yield) (MPa) for three average gradients, (a) 43°C/cm, (b) 14°C/cm, (c) 7°C/cm. Radius 40 mm, length 80 mm, boron oxide thickness 50 mm 224 Fig. 9.42 MRSS-CRSS (Yield) (MPa) for four temperature profiles with average gradients, (a) 33 C/cm, (b) 17°C/cm, (c) 11 C/cm and (d) 8°C/cm. Radius 27.5 mm, length 55 mm. Encapsulant thickness 21 mm. Argon pressure 2 atm 229 Fig. 9.43 Correlation between the measured temperature gradients across the encapsulant and the heat transfer coefficient values as a function of gas pressure and gas nature. Curve A from Refs. 143-144 using convection from a vertical wall ; and curve B using convection from an horizontal surface 234 Fig. 9.44 Element and nodal configuration at the interface for crystals with curved interface, (a) Convex interface, (b) Concave interface 237 Fig. 9.45 Fig. 9.46 Fig. 9.47 Fig. 9.48 Fig. 9.49 Fig. 9.50 Fig. 9.51 Fig. 9.52 Fig. 9.53 XXI Temperature (10 C) and Von Nises stress (MPa) fields for a crystal with a convex interface shape. Radius 27.5 mm. Length 27.5 mm. Encapsulant thickness 21 mm. Temperature profile as shown in Figure 8.19, curve 4 239 MRSS (MPa) contours for a crystal with convex interface at three lengths, (a) 27.5 mm, (b) 55 mm, (c) 82.5 mm. The largest stresses at the interface edge are (a) 7.22 MPa, (b) 9.19 MPa, (c) 9.66 MPa. Radius 27.5 mm, Boron oxide thickness 21mm. Profile as in. Figure 8.19, curve 4 240 MRSS-CRSS (Yield) (MPa) contours for a crystal with convex interface at three lengths (a) 27.5 mm, (b) 55 mm, (c) 82.5 mm. Radius 27.5 mm, boron oxide thickness 21 mm. Profile as in Figure 8.19, curve 4 242 MRSS-CRSS (Yield) (MPa) contours in a (001) plane at a distance of 11 mm from the cone in the crystal shown in Figure 9.47(a) 244 MRSS-CRSS (Yield) (MPa) contours in (001) planes at three distances form the cone. (a) 79.75 mm, (b) 66.0 mm, (c) 52.25 mm. Section corresponds to the crystal shown in Figure 9.47(c) 248 (a) MRSS (MP) and (b) MRSS-CRSS (Yield) (MPa) for a crystal with convex interface. Boron oxide thickness 50 mm. Radius 27.5 mm. Length 55 mm. Average gradient 50°c/cm 249 (a) MRSS (MP) and (b) MRSS-CRSS (Yield) (MPa) for a crystal with convex interface. Boron oxide thickness 50 mm. Radius 27.5 mm. Length 55 mm. Average gradient 17°C/cm 250 (a) MRSS (MP) and (b) MRSS-CRSS (Yield) (MPa) for a crystal with convex interface. Boron oxide thickness 21 mm. Radius 40 mm. Length 80 mm. Average gradient 55°C/cm 252 (a) MRSS (MP) and (b) MRSS-CRSS (Yield) (MPa) for a crystal with convex interface. (a) Radius XXI I 27.5 mm, (b) 40 mm. Boron oxide thickness 21 mm 254 Fig. 9.54 MRSS-CRSS (Yield) (MPa) contours in a (001) plane at a distance of 2.75 mm from the interface in the crystal shown in Figure 9.53(a) 256 Fig. 10.1 Temperature (103°C) field during cooling at four times. (a) 5 s, (b) 10 s, (c) 20 s, (d) 60 s, Initial gradient close to interface 70 C/cm. Argon temperature 1000°C 260 Fig. 10.2 MRSS-CRSS (Yield ) (MPa) during cooling for the four temperature fields shown in Figure 10.1(a-d). (a) 5 s, (b£ 10 s, (c) 20 s, (d) 60 s. Initial gradient 70 C/cm. Argon temperature 1000 C 263 Fig. 10.3 MRSS-CRSS (Yield ) (MPa) contours in (001) planes at four distances from the bottom (interface) in the crystal shown in Figure 10.2(b) 10 s, (a) 2.75 mm, (b) 8.25 mm, (c) 16.5 mm, (d) 27.5 mm 268 Fig. 10.4 MRSS-CRSS (Yield ) (MPa) in (001) planes at two distances from the bottom in the crystal shown in Figure 10.2 (d) 60 s, (a) 2.75 mm ( b) 8.25 mm 271 Fig. 10.5 (a) Temperature field (103°c) and (b) MRSS-CRSS (Yield ) (MPa^ field for a crystal cooling in argon at 1000 C after 10 s. Initial gradient 35 C/cm 272 Fig. 10.6 (a) Temperature field (103oC) and (b) MRSS-CRSS (Yield ) (MPa^ field for a crystal cooling in argon at 1000 C after 60 s. Initial gradient 35 C/cm 273 Fig. 10.7 MRSS-CRSS (Yield ) (MPa) in (001) planes at two distances from the bottom in the crystal shown in Figure 10.5 (b) 10 s. (a) 8.25 mm (b) 27.5 mm 275 Fig. 10.8 (a) Temperature field (103oC) and (b) MRSS-CRSS XXI11 (Yield ) (MPa^ field for a crystal cooling in argon at 1000 C after 10 s. Initial gradient 17.5°C/cm 277 Fig. 10.9 MRSS-CRSS (Yield) (MPa) in (001) planes a distance of 27.5 mm from the bottom in the crystal shown in Figure 10.8 (d) 10 s 279 Fig. 10.10 (a) Temperature field (103oC) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in boron oxide at 1000 C after 10 s. Initial gradient OoC/cm 280 Fig. 10.11 (a) Temperature field (103°C) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in boron oxige at 1000°C after 10 s. Initial gradient 5oC/cm 281 Fig. 10.12 (a) Temperature field (10 C) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in boron oxide at 1000°C after 10 s. Initial gradient 17.5°C/cm 282 Fig. 10.13 (a) Temperature field (103oC) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in argon at 800 C after 10 s. Initial gradient 70 C/cm 284 Fig. 10.14 (a) Temperature field (103oC) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in argon at 800 C after 10 s. Initial gradient 35 C/cm . . 285 Fig. 10.15 (a) Temperature field (103oC) and (b) MRSS-CRSS (Yield ) (^Pa) field for a crystal cooling in argon at 8 OoC after 10 s. Initial gradient 17.5°C/cm 286 Fig. 10.16 Temperature grofiles in the crystal cooling in argon at 800 C for two initial gradients in the crystal. (a) 70°C/cm (b) 35°C/cm. The right part corresponds to the radial profiles and the left part to the axial profile. Full and broken lines correspond to the axis and surface temperatures respectively 288 XXIV Fig. 10.17 Temperature profiles in the crystal cooling in boron oxide at 1000 C for two initial gradients in the crystal. (a) 70°C/cm (b) 35°C/cm. The right part corresponds to the radial profiles and the left part to the axial profile. Full and broken lines correspond to the axis and surface temperatures respectively 289 XXV LIST OF TABLES Table 5.1 Observed (Ng ) and Calculated (N and N ) Dislocation nsities for Gallium Arseniae Single Crystals 39 Table 7.1 Resolved Shear Stress Components in (001) Crystals 65 Table 7.2 Resolved Shear Stress Components in a (010) Plane for a [001] Crystal 67 Table 8.1 Values of physical parameters used in the calculations 92 Table 9.1 Effect of Cone Angle on Thermal and Stress Fields 124 Table 9.2 Effect of Ambient Temperature on Axial Temperature Gradient and Stress 150 Table 9.3 Effect of Non-linearity of the Ambient Temperature Profile on Stress 162 Table 9.4 Table of Symmetry Correlations 198 Table 9.5 Effect of Thermal Conditions on Stresses 226 Table VI.1 Rotation performed for the calculation of the RSS 357 Table VI.2 Resolved Shear Stress component in the <110> {111} slip 358 Table VI.3 Compact form of the RSS from Table VI.2 360 XXVI LIST OF SYMBOLS ( e ) [B]* ' Matrix of derivatives of the interpolation functions i 1 (e ) {F }. Initial force vector at node i in element e, 2x1. o i 4x2 [B']|e) Non-dimensional [B]je) matrix, 4x2. [C] Compliance matrix, 4x4. [C1] Non-dimensional [C] matrix, 4x4. ~ (e ) {d} Displacement vector in element e, 2 x 1. (e) {d} Nodal displacement vector in element e, 2n x 1. (e ) {d}* Nodal displacement vector at node i element e, 2 x 1 *d*red Reduced {d}*e) vector, 2x1. {d1} Global displacement vector in reduced form, 2n x 1. dV Elemental volume. E Young's Modulus, MPa. (e) {F}v Force vector for element e, 2n x 1 . ( e ) {F}. Force vector at node i in element e, 2 x 1. (e) (e) {F '}v ; Non-dimensional {F }. vector, 2x1. o o 1 i (e) F 1 Element of {F '} ' vector, o . o l {F '} Non-dimensional global force vector, 2n x 1. o g Number of nodes in element h Heat transfer coefficient (h.t.c.) h Convective h.t.c. c h Radiative h.t.c. r (e ) [k] Stiffness matrix for element e, 2n x 2n. (e) [k].. Stiffness submatrix for nodes i and j in element e, i J 2x2. XXV11 [k']*j* Non-dimensional (k}!^ matrix, 2x2. k^j Element of {M^* matrix. [K1] Non-dimensional global stiffness matrix. (K } Force vector for temperature field calculations. [KH] Stiffness matrix from boundary conditions in temper ature field ca1cua1tions. [K^] Stiffness matrix for temperature field calculations L Differential operator. Natural co-ordinate. ljj Length of element at thne boundary. m Degree of freedom in element. ->-n Unit normal vector to a surface. N Total number of nodes in the crystal. [N] Interpolation functions vector. r Radial co-ordinate. r Cristal radius. o t Time. * t non-dimensional time. T Temperature, °C. TQ Reference temperature. T„_ Temperature at melting point. MP Uj Radial displacement of node i. u(r,z) Radial displacement at point r, z. v Growth velocity. Volume of element e. -(e) Wj Axial displacement of node i. w. (r,z) Axial displacement at point of r, z. Weighting function. Axial co-ordinate. Coefficient of thermal expansion, °C 1. Non-dimensional temperature. Ambient non-dimensional temperature. Surface non-dimensional temperature. Eigenvalues of Fourier type algebraic equation Shear strain component in {£} tensor. Area of element (cross-section). Strain tensor. Initial strain tensor. Radial strain component. Azimuthal strain component. Axial strain component. Non-dimensional axial displacement. Eingenvalues of Bessel type algebraic equation Thermal diffusivity. Poisson's ratio. Non-dimensional radial displacement. Stress tensor. Radial stress component. Azimuthal stress component. Axial stress component. Von Mises stress, MPa. Principal stresses. Non-dimentional radial stress component. Non-dimentional azimuthal stress component. Non-dimentional axial stress component. Stress tensor at node i, element e. Shear stress component. Non-dlmentional shear stress component. XXX ACKNOWLEDGEMENTS I would like to thank Dr. Fred Weinberg and Indira V. Samarasekera for their assistance and valuable discussions during the course of this work. Support form the NSERC (Canada), Cominco Ltd., (Canada) and Universidad Nacional de Misiones, (Argentina) is gratefully acknowledged. 1 CHAPTER 1 INTRODUCTION Gallium Arsenide (GaAs) is a crystal semiconductor compound that is increasingly being used in the production of semiconductor devices particularly microwave and microelectronic . . 1-3 devices The advantages of GaAs over silicon (Si) are due to the differences in physical properties between both materials. Electron mobilities in GaAs are about four times higher than in Si with a power consumption which is a tenth of that in silicon. These become particularly significant as the VLSI circuits become more complex and the operating speed increases. In 1974 the first operating GaAs gate was produced. Ten years later GaAs IC's have 3 been produced with 10,000 gates or 4 SKRAM . In addition IC's are becoming smaller. A 1.1 by 1.6 mm chip may contain more than 600 active components plus contact pads, and work at about 10 4 Gbit/sec . These facts indicate that GaAs is the most promising 5 material for the next generation of supercomputers In optoelectronic devices Si has fewer applications than GaAs. Si does not glow brightly enough and does not lase because of the band gap structure On the other hand GaAs LED1 s are bright, can be made of different colours by impurity additions, and GaAs lases well because of its direct band gap structure1. At present only about 11 * of the GaAs chips have optoelectronic applications. By 1992 the percentage Is expected to Increase to 7 25 % . The combination of micro and optoelectronic properties of GaAs has led to the development of monolithic optoelectronic 8 -10 integration of simple circuits . Special emphasis has been placed on the development of the multiquantum well laser. In the area of solar cells GaAs is an excellent photoconverter with an efficiency of 20-24 %, as compared to Si at 15-18 % . GaAs solar cells are expected to reach efficiencies of 30 % with monolithic or hybrid tandem cells which are presently fabricated* * . GaAs devices have advantages in radiation fields and at 4 higher temperatures. They can operate in radiation fields 10 times larger that Si devices without deterioration, and at higher temperatures than silicon. One of the disadvantages of GaAs at present is its high cost 2 2 compared to Si (5.6 cents/mm for GaAs compared to 0.16 cents/mm 1 2 for Si ). However after all fabricating costs are considered for devices, the difference in wafer cost represents only a small fraction of the total and is therefore proportionately less 7 significant . One disadvantage of GaAs, at present is its high dislocation density, as compared to silicon which is dislocation free. Dislocation in many cases may affect both the yield of devices during fabrication and the performance of the devices. 3 This investigation focuses on the dislocations present in melt grown GaAs. The dislocations are believed to be generated during liquid encapsulated Czochralski growth (LEC) by thermal stresses in the crystal. A mathematical model will be developed to determine the thermal and stress fields in the growing crystal as a function of the many growth variables. The objective is then to use this model to control and modify the growth conditions and thus reduce the dislocation density in a controlled way. 4 CHAPTER 2 THE GROWTH OF BULK GaAs CRYSTALS 2 . 1 The Liquid Encapsulated Czochralski Technque (LEC) 13-15 In the LEC technique , shown schematically in Figure 2.1, a single crystal of controlled orientation is pulled from the melt, vertically upwards, starting from a seed crystal. Both the seed and the melt are rotated at different velocities and in directions counter to each other. In general high purity Ga and As are placed in the melt crucible and the intermetal 1 i c GaAs synthesized in the high pressure crystal puller. Ga melts at 20.8°C. As melts at 817°C at 28 atmospheres. The melting temperature of GaAs is 1238°C. At this temperature the arsenic partial pressure is 0.98 atm. Pressures of 50 atm. are therefore required to synthesize GaAs. Once the GaAs has formed, a much lower pressure of several atmospheres may be used to keep the GaAs from decomposing. Commercial crystal growers, in particular the "Melbourn" grower from Cambridge Instruments Ltd. is used extensively to grow GaAs crystals. Pressures inside the chamber can be raised to 50-70 atm. as required. In addition to the pressure inside the growing chamber, it was found that liquid encapsulation of the melt and the solid at the highest temperature was necessary to maintain stoichiometry. This is done by encapsulating the liquid with B 0_ which also covers the lower 2 o end of the growing crystal. The melt and encapsulant are le Figure 2.1 Schemat i c of a LEC pulling chamber 1 fi contained in a quartz or a pyrolitic boron nitride crucible . Using the high pressure (HP) LEC process in the Melbourn puller, controlled orientation <100> crystals up to 75 mm in diameter and 8 kgm in weight can be grown with dislocation densities in the order of 104 / cm2 As an alternative to the high pressure crystal growers, single crystals are being successfully grown at 1-2 atmospheres pressure using the LEC technique (LPLEC). In this case the starting material is polycrystal1ine GaAs which has been synthesized separately in a pressure vessel. A novel 4 17 technique ' permits in-situ synthesis in the LP growers by injection of As through the BO layer into a Ga melt contained in the crucible. With the LPLEC growers, crystals of 100 mm 18 — 20 diameter and 14 kgm weight have been produced The crystal growth process starts with a seed crystal, usually oriented for <100> or <111> growth, which is dipped into the melt. The crystal is then necked to reduce the propagation of 2 1 dislocations from the seed to the crystal and the diameter then slowly increased to the full diameter of the crystal. The growth is monitored visually and with a TV camera. Once the diameter of the crystal is established by manual control of the power input, further control is done with an automatic weighing system which adjusts the power input with computer 2 2-24 control Human errors, inabilities to see clearly through the encapsulant, and systematic errors in the weighing procedure 2 5 due to capillary forces , result in variations in the crystal 7 diameter of the order of 5 mm. Attempts are being made to reduce these problems6'19'26. When crystal growth- is completed, the crystal is removed from the ^z°3 and slowly cooled in the chamber to minimize thermal stresses. GaAs is normally semi-insulating. In some cases i soe 1 ec t r on i c additions, such as In, are added to the melt to improve properties. Other solute additions are made to the melt to make the material n- or p-type semiconductors. For n-type material dopants such as S, Se, Te, Sn, Si and Ge 2 7 are added. For p-type material Zn and Mg are added. The effect of solute additions and impurities on the electrical and mechanical properties of LEC-grown GaAs crystals and devices are discussed below. 2.2 Classification of Defects in LEC grown GaAs crystal A defect is defined as any deviation from the crystal structure. Defects consist of : i. Dislocations ii. Chemical impurities iii. Deviation from stoichiometry iv. Lattice defects, 1. vacancies and interstitials ; 2. subgrains, grain boundaries, stacking faults, striations and others ; 8 3. inclusions and precipitates ; 4. microdefects, consisting of small dislocation loops, precipitates, helicoidal dislocations and others. These defects have characteristic lengths of 100 to 1 nm28"30. The origin and nature of many of these defects is not 30 31 clearly understood ' . Generally a number of defects are presented in the crystal, which interact with each other and multiply. An example is the presence of small precipitates on 3 2 dislocation lines . The precipitates are believed to be crystallites of As embedded in Ga matrices resulting from 1) direct condensation of As point defects or 2) sweeping up of existing precipitates by moving dislocations. 2 . 3 Dislocations in LEC-GaAs Large diameter GaAs <100> single crystals produced by a 4 Melbourn puller have a dislocation density of approximately 5x10 2 3 3 /cm . Considerable effort is being directed toward producing low or zero density large diameter GaAs crystals. In 1982 the largest undoped GaAs crystal grown free of dislocation was 15 mm34, in 1985 a crystal 50mm in diameter was reported to have been grown dislocation and striation free ' . In the latter case the crystal was grown using a modified LEC technique called VMFEC which includes a vertical magnetic field. The crystal in this case is fully encapsulated and the melt is doped. The dislocation density can be reduced to 10° /cm by doping the melt with n, p or isoe 1 ectronic Impurities. Recent improvements of the thermal conditions during growth have resulted in undoped 3 2 crystals with etch pit densities (EPD) of 5x10 /cm for 5 cm 3 7-39 diameter crystal . In one case 70 mm <100> undoped crystals were grown in low vertical temperature gradients having EPD of 3 2 5x10 /cm over 70 % of the wafer area and throughout 75 % of the 4 0 ingot length Typical dislocation distributions on transverse sections of GaAs crystal are shown in Figure 2.2. In Figure 2.2a41 the dislocation density (EDP) is shown along <100> and <110> directions on a (100) GaAs wafer surface showing maximum dislocations at the outside and minimum midway between the outside and the center, giving a W shape distribution across the entire wafer. Figure 2.2b shows EPD distribution in a (100) wafer surface taken near the tail end of a 5 cm diameter crystal. The etch pit distribution has the four-fold symmetry normally observed, with higher densities at the outside and centre of the wafer. Close to the end of the crystal, EPD maps on 5.0 cm diameter crystal show the four-fold symmetry degenerates to two fold symmetry. This distortion is caused by the shift of the two minima in the [110] and [110] directions to positions closer to 3 3 the edge of the wafer 10 C I 10] C 1003 b 2.2 a) Experimentally determined41 dislocation density along [110] and [010] . b) EPD map33 on a 5cm dia. wafer from the tail end . 104 Etch Pits/cm . 1 < yellow < 12 < green < 15 < blue < 20 < black < 30 . 11 2. 4 Effect of Dislocations on Properties of GaAs Devices Dislocations are known to affect the electronic properties of GaAs in the following ways 1) They are considered to be 4 2 non-radiative recombination centers ; 2) They absorb non A Q radiative impurity atoms or point defects ; 3) When dislocations are arrayed in cellular structures, carriers are not 4 3 active at the cell walls The effect of dislocations on the properties of microelectronic and optoelectronic devices depends on the nature of the device and how it is fabricated. In optoelectronic devices, it has been reported that dislocations seriously reduce 4 4  2 the lifetime of lasers , reduce the efficiency of LED1s , and under an applied stress, increase the degradation of diodes by one order of magnitude45,46. The mechanism for these effects is 47 unclear With microelectronic devices it was considered that A Q dislocations did not significantly affect their properties However at the present time this has changed and dislocations are believed to affect the electrical properties of IC's of GaAs. These results are presented and discussed in Appendix I. 12 CHAPTER 3 THE ORIGIN OF DISLOCATIONS IN LEC GaAs CRYSTALS Dislocations in GaAs are either seed related or growth related. The seed related dislocations result from the propagation and multiplication of dislocations from the seed into the main crystal. These dislocations can be reduced or eliminated by necking the seed crystal, as is commonly done, or by starting 3 5 with a dislocation free seed crystal The growth related dislocations in the crystal are generally attributed to the thermal stresses generated in the crystal during growth. The dislocations nucleate in regions of maximum stress near the crystal surface, then propagate along slip planes and multiply. The dislocations can also be generated by thermal stresses at precipitates or foreign particles, and by the condensation of vacancies into dislocation loops which glide. Examination of a <100> melt grown GaAs° crystal wafer by transmission X-ray topography of etch pits (EPD) shows that the dislocations are not uniformly distributed throughout the wafer as shown in Figures 3.1 a at C . As described previously the dislocation density along a crystal diameter follows a W pattern with higher concentrations at the outside and center. In addition the dislocations form cells by concentrating along cell walls. Cell diameters have been observed of 0.1 mm in areas of 13 Figure 3.1 Transmission X-ray topography in GaAs - (a^^and (b) undoped ; (c) Te-doped, (110) axial section 14 5 2 dislocation densities of 10 /cm (based on EPD). At lower densities the cells can increase in size up to 3 mm30,110. The cell wall directions on a (100) crystal face are randomly oriented. The formation of cells in the GaAs wafers is not clearly understood. The most favoured mechanism is that dislocations generated near the crystal walls and in the centre glide and climb after solidification eventually forming the cell structure to minimize their energy. An alternative mechanism has been proposed by Holmes111 in which the cell structure is attributed to constitutional supercooling at the surface, similar to cell 112 formation in metal alloys. This mechanism appears unlikely because of the low impurity levels in the GaAs, high stoichiometry at the interface and slow growth velocities. In addition to cell structures on GaAs crystals, long linear arrays of dislocations are also observed as shown in Figure 3.1b at L. The linear arrays are observed in areas having a 4 2 dislocation density of less than 2x10 /cm and produce a small tilt orientation difference of about 0.01° in the crystal across the line. These arrays are roughly similar to lineage sub-boundaries in melt grown metal crystals attributed to the condensation of vacancies immediately behind the interface, forming dislocation loops which climb to form lineage boundaries. Slip lines are sometimes observed on etched GaAs wafers as shown in Figure 3.1a at S, generally in areas of lower dislocation density (103 /cm2)113. 15 Specific observations and discussions of dislocation generation and multiplication in GaAs are given below : 1. The generation of dislocations by thermal stresses has 114 been reported by Milvidskii and co-workers and others11^*'11*5. Thermal stresses in the growing crystal result from radial and axial temperature gradients. When the local stress is greater than the critical resolved shear stress, glide occurs on the _ 117 118 (111) slip planes in <110> directions ' . This 119 mechanism was proposed for Si by Billig and was 12 0 applied to Ge by Penning 2. The most probable mechanism for the generation of dislocations is stress concentration at heterogeneous sources at the crystal surface or in the bulk material. Small precipitates of As have been found at 3 2 dislocations Amorphous precipitates of Ga or As have 3 0 been observed on the cell walls in undoped GaAs Insular polycrystal1ine GaAs having diameters of 1000°A have been observed in Si doped GaAs which are 30 associated with dislocation generation . Steinemann 121 and Zimmerly associate dislocation generation with precipitates of excess Ge or impurity atoms. Excess Ga vacancies11^'122 in particular and native point defects 123125 in general which precipitate into dislocation loops have been considered as a source of dislocations. 12 6 12 7 Lagowski et al. ' have proposed a theory in which 16 the Ga vacancy Is the critical native defect in the formation of the dislocation loops. This theory is presented in part b of Appendix I. o 3. The stress distribution in growing crystals has been 12 8 considered in several ways. Nikitenko and Indenbom used photoelastic methods to observe the stress distribution in crystals. They also used mathematical models to establish the stress field in a growing crystal and derived the dislocation distribution based on local stress levels above the critical resolved shear stress. The distribution they predicted agreed reasonably well with observations. The correlation of thermal stress fields to dislocations has also been observed using X-ray diffraction to measure lattice strains in GaAs129-131, photoelastic methods132,133 and photo1uminescent scanning134. In summary, several causes for the origin of dislocations in GaAs have been proposed in the literature. In crystals grown from melts far from stoichiometry the Ga vacancy may control the formation of dislocations. Under normal growth conditions, however stresses play a fundamental role in controlling dislocation densities by a glide mechanism. More experimental evidence supporting the stress related mechanism is presented in the next chapter. 17 CHAPTER 4 DISLOCATION DENSITY AND CRYSTAL GROWTH The density and distribution of dislocations in LEC grown GaAs crystals is strongly dependent on a number of growth parameters. The dislocation density may be reduced, improving the crystal quality, by changing the growth conditions : 1) Reducing the radial and axial thermal gradients during growth 2) Adding impurities to solution harden the crystal increasing the CRSS 3) Reducing variations in the crystal radius during growth 4) Maintaining stoichiometry of the Ga and As in the melt. More uniform dopant and impurity distribution in the crystal may be obtained by stopping fluid flow in the melt due to convect i on. 4•1 Stresses in the Crystal Due to Thermal Gradients Thermal stresses in the crystal in LEC GaAs growth depend primarily on 1) the crystal dimensions and 2) the ambient variables affecting the temperature distribution in the crystal. 18 4.1.1 Crystal Dimensions In this category the variables are crystal diameter, crystal length, cone angle and neck and seed dimensions. Increasing the diameter of the crystal generally increases the dislocation density. At very large diameters this may not be the case. Increasing the diameter of crystals from 5.0 - 7.5 cm diameter to 12.7 - 15.2 cm diameter did not change the 1 8 dislocation density The increase in dislocation density with increasing diameter can be due to an increase in axial and/or radial thermal 13 5 13 6 gradients. Experimental observations are contradictory Results of calculations are also contradictory. According to 13 7 Jordan et al. both axial and radial gradients increase with 13 5 radius. On the other hand, Brice and Buckley-Golden and 13 8 Humphreys report that axial temperature gradients and axial temperature curvatures decrease with increasing radii. The EPD indicating dislocation density, increases with distance from the seed end of the crystal. The dislocation distribution across the crystal has a W shape, as shown in Figure 4.1., which is maintained throughout the crystal. In low 113 dislocation density crystals Elliot et al. have reported high EPD values at both the seed and tail ends of the crystal. The effect of cone angle on EPD has been evaluated experimentally by Chen and Holmes110. They found that the EPD decreases with increasing cone angle up to a value of about 19 Figure 4.1 Radial dislocation density profiles across wafers obtained from the front, middle, and tail of a crystal. The radial profiles are "W" shaped, and the average EPD increases from the front to the tail. 20 25 degrees. Above this value they found no correlation between 3 9 the EPD and cone angle. This agrees with Watanabe et al. Thermal stresses depend on the elastic constants of GaAs. Accordingly the crystal orientation should affect the dislocation 1 1 R density. For horizontal Bridgman growth of GaAs Plasket et al. have found that the lowest dislocation density was obtained for crystals grown in the <013> direction. This cannot be taken advantage of because circular GaAs crystal wafers having (100) surfaces are required for production of IC's. 4.1.2 Thermal Conditions During Crystal Growth Thermal gradients in the crystal depend directly on the thermal conditions in the surrounding environment through heat transfer at the boundary of the crystal. The heat transfer process can be characterized in a simple way by heat transfer coefficients and the ambient temperature around the crystal. The main factors affecting the heat transfer are : gas pressure, boron oxide layer thickness, and the geometry and position of the heater and shielding system. Increasing gas pressure increases the EPD39'110 ' 11^ . This effect is explained in terms of the variation of the heat transfer coefficient110, which according to calculations 13 7 13 9 increases as the square root of the pressure ' The change in heat transfer coefficient between the crystal and gas cannot explain the variation in the EPD observed. 21 The gas pressure during crystal growth cannot be reduced to reduce dislocation density. Without sufficient pressure at high temperatures As atoms leave the crystal. Excess Ga on the surface will result in pitted surfaces, twins and polycrystalinity 110,140,141 jn addition, changing the gas pressure changes the atomic fraction of As in the melt. This can influence the 14 2 electrical resistivity of the material The dislocation density is also affected by the type of inert gas used. A nitrogen atmosphere gives an EPD one order of 3 9 magnitude higher than argon . With other inert atmospheres the EPD counts were found to increase in the order Kr, Ar, and 143 144 He ' The effect is attributed to the different heat transfer efficiencies of the gases which change the thermal gradient in the boron oxide encapsulant. In a conventional Melbourn grower increasing the BO layer decreases the dislocation density in the crystal. Doubling the encapsulant thickness from 15 to 30 mm reduces by half the axial , . ,.145-147 . . * c« , J35,148 gradient . At present 50 mm thick layers are used Enhanced heating of the B2°3 layer through windows in the susceptor cylinder results in lower axial gradients and therefore lower EPD. The addition of baffles above the crucible to reduce convection of the gas reduces the axial and radial gradients to 15°C/cm and 3°C/cm respectively for a Bo0- thickness of 30 mm. With this technique 5 cm diameter crystals can be produced with 3 2 dislocation densities of 10 /cm in large areas. 22 4.2 Alloy Addition to the GaAs The reduction or elimination of dislocations in GaAs by 14 9 alloy additions to the melt was reported as early as 1972 . The effectiveness of specific additions in reducing the as grown dislocation density in GaAs (20-25 mm diameter) is shown in 150 Figure 4.2 . It can be seen that Te is the most effective in reducing the dislocation density followed by In, Sn and Zn. Seki 151 and coworkers determined that Zn, S, Te, Al and N are 15 2 15 3 effective in reducing the dislocation density. Silicon ' and 154 Boron have also been shown to have the same effect. Dislocation free GaAs has been obtained doping the melt with 15 5 N (20 mm in diameter) , Te (22 mm in diameter, <111> crystal)156, Si (40-50 mm diameter)157, In (50 mm in ,. . ,35,36,158 diameter) One important aspect of doping is that the addition produces n- or p-type material. If semi-insulating (Si) GaAs is required, only isoelectronic impurities can be added. Indium is the promising impurity to decrease dislocation density and obtain . . , ^. . ,159-162 semi-insulating material The primary effect of doping is to increase the critical stress for dislocation generation. For undoped GaAs the CRSS is 18 3 7 g/mm while for Te-doped GaAs (2x10 atoms/cm ) the CRSS is 16 3 16 3 40 g/mm . This effect, as explained by Milvidskii et al . , appears because high temperature and low stresses create 23 favourable conditions for the formation of a stable impurity atmosphere (Cottrell atmospheres) surrounding moving C [atom- cnr3J Figure 4.2 Mean density of "grown-in" dislocations in GaAs single crystals (20-25 mm in diameter) grown by the Czochralski LEC Technique, as a function of dopant concentration : (1) Te, (2) Sn, (3) In, (4) Zn. dislocations. This reduces effectiveness of an element in 2 given by a factor U /D where atoms with dislocations and coef f i cient. the dislocation velocity. The stopping dislocation movement is U is the elastic interaction of D is the impurity diffusion 24 A number of specific mechanisms have been proposed to account for the effect of solute atoms on dislocation mobility. 151 Seki et al. associated dislocation movement with breaking of bonds between solute elements and the matrix, with limited 164 success. Sher et al. related the bond length (d) to dislocation energies and hardness in semiconductors. They showed that dislocation energies per unit length are proportional to -3 -9 -5 -11 d - d and hardness to d - d This analysis could be used to account for the reduction in dislocation density in GaAs 165 151155 by adding BAs and GaN ' and predicts GaP additions will reduce the density. The reduction in dislocation density by In can be accounted for by solution hardening166. In this case the solute entity is not the In atom alone but consists of a cluster of five atoms. The cluster is formed by one In atom bounded by four As atoms, InAs4> in a tetrahedral configuration which is embedded in a GaAs matrix. These results are supported by the experimental observations.that the fee (Ga.In) sublattice parameter increases linearly with In concentration and that the Ga-As and In-As bond length remains roughly constant as x is varied in Ga'1 X^InXAs16^. The dilation of an InAs4 tetrahedra as compared to a GaAs^ tetrahedra is 21 % . This is semi quantitative ly calculated to have a stronger effect than solution hardening in metals. Solute additions to the GaAs melt can markedly influence the behaviour of devices fabricated on the crystal wafer. 25 1) If sufficient solute is added constitutional supercooling could develop at the interface producing solute microsegregation in the crystal 2) Lineage structure can be developed in the crystal producing orientation variations. 3) Microdefects, including precipitates can be present in the crystal due to the solute present. As an example, Te additions at low levels produce stacking faults in the 168-173 crystal . Te additions at high levels produce prismatic dislocations in addition to stacking 172173 168 faults ' . S additions have the same effect Helicoidal dislocations were observed in GaAs with small 18 3 addition of Te (2x10 atoms/cm ) and with Sn, S and Zn , . . . . 168,174 additions Controlled additions of elements to GaAs are made to produce p- or n-type crystals. The elements selected should result in the desired type of crystal with minimal other defects produced, as well as the reduction in dislocation density. It has been shown that Si is the best dopant for GaAs to produce n-type behaviour, 18 3 175176 at levels of near 10 atoms/cm , as compared to Te and S 3 2 In this case dislocations densities of 3x10 /cm are obtained with no microdefects. At higher dopant levels, above 18 3 3x10 atoms/cm , straight and helicoidal dislocations become pronounced. Similar results are obtained with In dopant for 19 3 levels greater than 2x10 atoms/cm . At high In levels 26 20 3 (10 atoms/cm ) three different regimes of dislocations have 17 7 been reported by Pichaud et al . as shown in Figure 4.3 . In the figure the In concentration is shown to increase with distance along the growth direction (pulling axis) which changes the dislocation density. Position along the pulling axis Figure 4.3 Hypothetical distribution of the dopant (In) along the pulling axis of the ingot. The indium concentra tion oscillates around a mean value which is proportional to the distance from the top of the ingot . At the seed end and middle of the crystal, dislocations are tangled with a banded periodical distribution normal to the growth direction. In the final third of the crystal the bands disappear and the dislocation density is reduced. The oscillations in the In levels shown schematically in the figure are associated with convective flow in the melt. 27 Striation or linear boundaries are observed in undoped GaAs 17 8 crystals . The striation density can be reduced by applying a 17 9 magnetic field at kilogauss levels to undoped GaAs , Se 179180 137 doped ' and In doped material. With a vertical magnetic field thermal oscillations in the melt due to convection can be reduced to less than 0.3°C 181 but not eliminated180. However, at this time it is not clear how the magnetic field and convective flow quantitatively change the crystal characteristic. Numerical 18 2 183 simulations and modelling indicate magnetic fields at kilogauss levels do not suppress convective flow. Recent results 184 show a 2 KGauss field can stabilize flow and thermal fields The influence of a magnetic field on the impurity distribution in grown crystals is complex. Carbon concentrations are decreased 181 and Chromium levels are increased Residual impurities are changed, as evidenced by a reduction in electrical resistivity of undoped crystals185 . The addition of In to GaAs to reduce dislocation densities without changing the electrical properties presents difficulties. Macrosegregation of the In along the crystal axis will occur as in Figure 4.3 . In addition the lattice spacing in GaAs changes with the In concentration which affects the quality of epitaxial layers deposited on the surface in device fabrication. Specifically, misfit dislocations are produced at the SI GaAs substrate and MBE deposited GaAs above a critical deposited layer thickness. The critical layer thickness decreases exponentially with increasing In concentration. For a significant 28 reduction of dislocations, In concentrations greater than 19 3 2 0 3 10 atoms/cm are required. At concentrations of 10 atoms/cm the critical thickness of deposited material is about 1 micron which is small. The effect of In alloying in FET characteristic has been 18 7 examined by Hunter et al. . Using direct ion implantation on In alloyed crystal wafer the FET's showed uniform properties along the wafer. Inhomogeneous FET's characteristic are expected from arrays fabricated on wafers with inhomogeneity in the resistivity. Non uniform resistivities along radial and axial directions are characteristic of In doped GaAs crystals with 18 8 dislocation or dislocation free material The uniformity in the FET properties obtained by Hunter et al . could be a result of a post solidification annealing process as a result of the low growth velocities and low thermal gradients during crystal growth. Bulk annealing has produced uniformity improvements in 18 8 10 3 undoped GaAs and In doped GaAs 29 CHAPTER 5 MODELS OF DISLOCATION GENERATION IN GaAs Mathematical models for dislocation generation by glide due to thermal stresses have been developed using different approaches for the calculation of the thermal field in the crystal as well as the thermoelastic stress field. 5.1 Stress Fields 1. The temperature field in the growing crystal has been 18 9 calculated numerically assuming the interface is a paraboloid of revolution. The boundary conditions were determined experimentally by measuring the near surface temperatures at a number of positions along the crystal. The thermoelastic stresses were calculated assuming a cylindrical crystal with flat faces and using a plain strain approximation. The results are shown in Figure 5.1 for a <111> grown crystal. The calculated isotherms and RSS are shown as a function of position in the crystal in (a) and (b) for two crystals of different dimensions. The number to the right of the temperatures gives the CRSS. It is apparent that there are a few areas where the RSS is less than the CRSS (shown by the dashed lines). The isotherms in parts of the figure are first concave following the interface shape and change to convex after the 1200°C isotherm. Growth conditions and crystal characteristics are not given. In parts (c) and (d) of the figure 30 0-V 0.8 rIK (I 0.1/ fi• g 9, 710 Fig. 5.1 (a) and (b) - Isotherms and shear stress topography in two gallium arsenide single crystal grown under different conditions. The figures next to the curves are x values (kg/mm ) corresponding to the constant-stress lines. The dashed curves outline regions in which the effective stresses are lower than the reduced yield stress. The figures in parentheses along the ordinate, axis gre the experimental critical stresses in 10 kg/mm . (c) and (d) - Distribution of shear stresses T., and T3 (c) and dislocation density (d) over the cross-section of a gallium arsenide single crystal. 31 the mean RSS shows the typical W- shaped distribution of the dislocations along the wafer diameter ; the stress symmetry is not obtained. The effect of growth conditions is not analysed. 2. The temperature field in growing GaAs has been determined analytically by Jordan et a 1 .137•148•190. They used a quasi-steady state approximation of the heat conduction equation in cylindrical coordinates. For the boundary conditions they assumed a constant temperature at a flat solid/liquid interface and convection/radiation heat loss at the surface of the crystal following Newton's Law of cooling. The heat transfer coefficients used were evaluated separately as a function of temper-13 9 191 ature ' . Using the calculated thermal fields, the stresses in the crystal were calculated using a plain strain approximation, and the RSS was determined for a <100> oriented crystal. The local dislocation density was assumed to be proportional to the total RSS. This was defined as the sum of the absolute values of RSS acting on {111} <110> slip system. The results are shown in Figure 5.2 in which the total RSS (TRSS) is plotted on a transverse plane of a cylindrical crystal. Typical four-fold symmetry is observed with stress minima midway between the outside and centre of the crystal along the <110> directions. The principal conclusions derived from the model follow : 1) Doubling the radius of the cylindrical crystal from 2 to 4 cm more than doubles the TRSS. 2) The dislocation density increases with increasing heat transfer coefficients. 32 3) The dislocation density can heat transfer coefficient is one order of magnitude small the calculations. be reduced to zero if the small enough, estimated as er than the values used in 3. The temperature field is calculated numerically and the thermal stresses determined. However, in this case stress relaxation during growth is included in the analysis (Vakhrameev 192 et al. ). The thermal field is determined by solving the steady state thermal conduction equation for the entire growth system including crystal, melt, gas and crucible. The encapsulant BO 19 3 is neglected The thermoelastic stress field is obtained by solving the displacement thermoelastic equation by a finite 19 4 difference method Results are shown in Figure 5.3(a-c). In Figure 5.3(a) and (b) the dislocation density is shown as a function of temperature for a Burger vector perpendicular to and inclined to the growth direction <111>. A considerable fraction of the dislocations is observed to be on the periphery of the crystal at temperatures well below the melting temperature. Figure 5.3(c) shows dislocation densities, thermoelastic stresses, relaxed stresses and CRSS for undoped GaAs along the axis of the crystal. Two well defined regions are observed where dislocations are formed, one near the crystallization front at 1238°C and the other near 1140°C. Thermoelastic stresses in this case are relaxed about 80 % . The growth conditions are not given and the effect of growth parameters is not discussed. 33 Figure 5.2 TRSS contours for the top wafer of a <001> GaAs boule. 34 a b c 5.3 Distribution of the calculated dislocation density in a gallium arsenide single crystal being grown in the <111> direction for slip systems with Burgers dislocation vectors perpendicular (a) and inclined (b) to the growth axis at the following temperatures: 1) 1234, 2) 1200, 3) 1140, 4) 1050 C - T. (c) Distribution of calculated dislocation density 1) thermoelastic stress, 2) relaxed thermal stress, 4) and critical stress for the formation of dislocation in undoped GaAs at the correspond ing temperature, 3) along the axis of a single crystal during its growth ; CF) crystallization front. 35 195 4. Dussaux has employed a finite element method to obtain both thermal and stress fields. The thermal field was calculated by solving the steady heat conduction equation in the crystal subject to Newton's Law of Cooling at the crystal surface as done . . , . . 137,148,190 by Jordan et al. The numerical scheme chosen permits more realistic boundary conditions to be used such as temperature gradients outside the crystal and temperature dependent heat transfer coefficients. In addition different geometries can be considered. The stress field is calculated using the minimum energy principle method and an axisymmetric field is assumed. From the stress components the Von Nises stress is derived. Stresses are not resolved in the slip system. The Von Mises stress contours in the crystal for a fixed boron oxide layer, and thermal gradients are presented for two crystal diameters and different crystal lengths. The model is employed to compare the LEC and LEK (liquid encapsulated Kryopoulos) methods and not to an extensive analysis of dislocation generation in LEC growth. 5. Thermal stresses have been calculated using analytical approaches in which the plain strain approximation is not assumed. Instead of plain strain an axisymmetric approximation is used. Qualitative good agreement with plain strain and FEM 19 6 calculations is obtained . Using the same axisymmetric 19 7 198 approximation ' , it was shown that the plain solutions are not always valid in determining the thermal streesses in the 36 crystal. These conclusions are obtained for semi-infinite geometries . 199 6. Galaktionov and Tropp have considered a non standard approach to introduce the effect of axial gradients. They assume that the axial gradient of the total stresses (transient and residual) are numerically equal to the thermoelastic stresses produced by a hypothetical temperature field given by the axial gradient of the real field. In equation form this is expressed as 3 9 0. . [T(x,y,z)] = O t T(x,y,z)] 3z 1J 1J 3z The method is used to calculate the residual stresses in infinite cylinders. 5 . 2 Calculation of Dislocation Densities The dislocation density at a point in a growing crystal is a function of the local stresses generated by the thermal strains. The density may be assumed to be proportional to the maximum RSS (MRSS) or to a lesser value if the stress is considered to be relieved by plastic flow. This has been considered in a number of different ways : 1. Jordan et al. (model 2) assumed that the thermoelastic stresses were partially relieved by plastic flow but made no estimate of the amount relieved. The flow was assumed to be proportional to the TRSS. 37 2. Vakhrameev et al. (model 3) calculated that 80 * of the thermoelastic stress Is relaxed by plastic flow. 3. Billig in 1956 and Indenbom in 1957 proposed that stresses are completely relaxed. 12 9 Billig estimated the dislocation density using the following simple analysis. Consider a thin slab of thickness 6r and length z. The application of a temperature gradient 6T/6r will expand the heated surface by 6r = az 6T. This expansion will bend the slab giving a curvature radius 1/R = (1/z)x(6z/6r). If stresses are relaxed by plastic deformation a number of edge dislocations are formed. The density n is given by the curvature and Burger's vector, n = 1/Rb, therefore n = (a/b) (6T/6r) Using this relation Billig estimated dislocation densities 4 2 in Ge of the order of 3.2x10 /cm for given growth conditions. Indenbom proposed a similar expression in which the axial temperature gradient is considered. This is given by n = (1/b) [d(aT) /dt] The above expression is part of a more complex equation that involves a tensorial dislocation density n, given by n = -grad T X a 38 where a is the tensor of the coefficients of linear thermal 2 0 2 2 0 3 expansion ' The contribution of the radial and axial temperature gradients to the total dislocation density is not , 119,204 clear Indenbom also proposed another expression to estimate the total dislocation density from plastic deformation e n = e/bL where L is the dislocation length which, for Ge and Si, is of the 10 6 2 0V order of 0.25 mm. The Indenbom equations were extented ' by introducing the CRSS in the glide plane to obtain the following expression for the dislocation density •+ _ n = (a/b)VTn - 2(T /Gb)(l/D) c r which reduces to the equation given by Indenbom when the CRSS is -y zero. In the equation VTn is the mean axial temperature gradient, D is the crystal diameter and G is the shear modulus. The dislocation densities calculated with this equation are compared to measurements on GaAs in Table 5.1. The columns headed Nl and N2 are calculated values for the first and second terms of the above equation. This equation is not used in the present investigation because it is inconsistent with the evaluation of dislocation densities from stresses in the crystal. This is because a constant axial thermal gradient does not produce stresses. 39 In summary from the reported results It is not clear to what extent the thermoelastic stresses are relieved. In deriving dislocation densities adjustable parameters are introduced in pseudo-empirical equations which are not tested under controlled laboratory conditions. In addition, in the equations in which dislocation densities are directly proportional to thermal gradients, the role of stresses in generating dislocations is not clear. Table 5.1 Observed (N ) and Calculated (N, and Nn) Dislocation exD 1* 2 Densities for Gallium Arsenide Single Crystals. VT N ' 3 6X0 2 |D,cm | | oil Ni.CB> I N9,cm | Growth method | deg/cm cm 1,0 142 5 103 1,7 104 2,2 10* 2,4 140 "3-5 10* 2,0 10* 2,2 10* 2,4 120 ~l-2 10* 1,7 10* 1,9 10* The Czochralski 2,4 120 "0,9 10* 1,7 10* 1,9 10* method 2,4 50 "5-6 103 5,7 10^ 7,8 10^ 2,7 190 5 10 2,7 10 2,9 10 0,8 450 "1 105 0,6 105 0,7 10^ 0,8 380 7-9 10* 5,3 10* 5,9 104 Free zonal 0,8 290 4-6 10 3,9 10 4,5 10 melting 0,8 210 2-3 10 2,6 10 3,2 10 Note and Ng were calculated with Eq.(2) and Indgnbomjs equation, respectively ; b = 4,10 cm |21] , a =6,2 10 deg and it was assumed that T /G - 1 10 . er 40 CHAPTER 6 OBJECTIVES In the previous chapters it has been pointed out that dislocation densities at the levels usually present in LEC GaAs are detrimental to the preformance of fabricated devices. New technological advances in the GaAs micro and optoelectronic industry require GaAs wafers of low dislocation density or free of dislocations. Dislocations in LEC GaAs are generated during growth mainly due to thermal stresses. On the basis of a glide mechanism thermoelastic models have been developed to describe the dislocation distribution in as grown crystals. These models employ analytical and numerical methods. The models can qualitatively derive the W-shape and four-fold symmetry of the dislocation distribution. However they cannot determine the axial distribution in the growing crystal, nor can they be used to develop changes in growth conditions to reduce or eliminate dislocations in GaAs. This results from the analytical solution limitations, both plain strain and axisymmetric approximations, and the limited scope of the Dusseaux numerical model. The models do not adequately take into account the primary growth parameters during crystal growth producing the thermal fields on which the models are based. Experimental measurements 41 of the thermal fields, as a function of the growth parameters, to incorporate into the model are very difficult to make in high pressure growers. The main objective of this investigation is to further develop the mathematical model for dislocation generation and multiplication taking into account all of the variables related to crystal growth. Experimental observations reported in the literature of thermal fields and physical quantities of the GaAs and BO, and made in a high pressure grower will be used to establish the boundary conditions in the model. The model will be able to determine the axial dislocation distribution and will have the flexibility to incorporate any change in the growth conditions . The model will be used to assess the dislocation density in crystals with the introduction of modifications of the thermal fields in the crystal growers. The objective would be to develop procedures for lowering or eliminating dislocation densities in the crystal without introducing other defects which affect the electrical behaviour of the crystals. The investigation was carried out in the following sequence : 1. Based on a glide mechanism due to thermal stresses, mathematical models for dislocation generation in LEC GaAs during growth and subsequent cooling, were formulated, developed and validated. 42 Using the models, most important factors affecting dislocation generation in GaAs crystals during and after growth, were determined. Based on the above results and the thermal history the role of thermal stresses in the generation of dislocations in LEC crystals, were clarified. 43 CHAPTER 7 FORMULATION OF THE MODELS The production of crystals can be divided into two parts crystal growth, which includes the solidification process, and the subsequent crystal cooling to ambient temperature. Both parts of the process are modelled separately. In the two models the basic assumption is made that the temperature fields and stress fields are decoupled2^®. A simplified flow chart of the models which derive the stress fields from the thermal fields is shown in Figure 7.1 The flow chart for both parts of the process are identical ; the difference between both models is the way the thermal fields are calculated. At a specific time during growth, the stress fields are derived from the calculated thermal fields. The stress components are then used to calculate the Von Mises Stress or the Resolved Shear Stress (RSS) at a given point in the crystal. The RSS are the projections of the stress tensor in specific planes and directions. In our case those planes and directions are given by the glide system of GaAs {111} <110>. For a crystal grown in the [001] direction, from the twelve combinations of planes and directions there are ten different RSS components. It is assumed in the model that when the maximum component of RSS is greater than a critical value, dislocations are generated and multiply during glide. The particular planes and directions in which the 44 TEMPERATURE FIELD STRESS FIELD VON MISES STRESS GLIDE MODE RESOLVE STRE ;D SHEAR ;ss MAXIMUM SHEAR S' RESOLVED rRESS NO DISLOCATIONS Figure 7.1 Flow chart of the model used to derive the stress fields. 45 RSS are maxima will determine the glide mode and crystallography of the dislocations being generated. It is also assumed that when one mode operates, the stress is relaxed and there is no further glide in other directions. 7.1 Model of crystal growth 7.1.1 Temperature Field 7.1.1.1 Governing Equations A complete analysis of the temperature field in a LEC-GaAs crystal during growth requires the solution of a Stefan's problem with complex boundary conditions. The exact solution of this problem is time dependent and includes the four phases, crystal, gas, encapsulant and melt as well as the mold and the heat input/output to the mold. The heat transfer mechanisms include conduction, radiation and convection. To treat this problem mathematically the following assumptions are made : 1. The temperature field in the crystal is axially symmetric, i.e. T = T(r,z,t) 2. The interface shape is unaltered by dissipation of latent heat and remains at the melting point, T M P 3. Heat transfer at the interface of the crystal (excluding the solidifying interface) follows Newton's Law of Cool ing 46 4. The heat transfer coefficient and ambient temperature are allowed to vary along the lateral surface of the crystal, i.e. in contact with the encapsulant or the gaseous atmosphere 5. The temperature field is changed by a change in crystal length with time, i.e. a quasi-steady state approxim ation. With the above assumptions the non- dimensional equation of heat conduction, with respect to a coordinate system fixed to the solidifying interface, as shown in Figure 7.2 , takes the form : 323 l 36 326 36 L (6) = -r- +  r- - 2V = o (a) 3p p 3p 3? 3£ (7.1) T - T MP 0 p = r/rQ , £ = z/rQ (b) where 2V = v rQ/ K , TQ is the reference temperature, rQ is the crystal radius, v the growth velocity and K the thermal conductivity. Solutions must also satisfy the following general boundary conditions : i. Dirichlet condition At the solid-liquid interface given by the suface of revolution f(r,z) = 0 , r < r^ 6 = 1 (7.2) lz=0 r melt Figure 7.2 Crystal configuration and coordinate the mathematical model. system used in 48 i i . Neumann condition At any point S on the cylindrical surface of the crystal, 33 33 (— nD + — nF)_ n = -h(z) r.rfi - 3nU)l (7.3) 9p P 3? £, S 0 S a where 3 (z) is the normalized temperature of the ambient adjacent to the crystal surface at a distance z from the interface and 3g is the normalized temperature of the crystal at that point ; h(z) is the heat transfer coefficient between crystal surface and ambient at the same point. The numerical scheme employed to solve this is given below. 7.1.1.2 Finite Element Equations Among the different approaches for obtaining the element equations in a finite element method, the Galerkin's method was selected. In this method, after the whole domain in which a solution for the field variable 3 is required is discretized in small elements, the element equation is obtained by performing the following integration over each element, S. . M3(e)) N <e) d D(e) D(e) i 0 with (e) [N] {3} (e) [H2 N0 . . N ] < g ^ g (7.4) 49 where the N's are the interpolation functions, the 3 ^ are the nodal field values in element (e) and g is the degree of freedom of the element . The interpolation functions must be such that there is continuity of the field variable and of its partial derivatives up to one order less than the highest order derivative in the operator. This means that interpolation functions for a C* problem are at first apparently required. However integration by parts and the use of the boundary conditions transform the problem in such a way as to show that less strict interpolation functions can be used. These calculations are presented in Appendix II where it is shown that the initial problem given by Equation (7.4) is transformed into the following equivalent equation 3N 3g(e) 3N. 33(6) 3g(e) ( + - + 2V N. ) dV(e) + D(e) 3P 9P 3£ 9? 1 35 . + r h(z) f N. 3(E) dS(e) = s(e) = rQh(z) B (z) f N.dS(6) (7.5) Js(e) where S^e' is the surface area of element (e) and dS is the elemental surface. It is now clear from the above equation that interpolation functions for C^0^ problems are required. 50 A suitable element shape chosen here for subdivision of axisymmetric bodies is the axisymmetric ring element with triangular cross-section. The field variable is assumed to vary linearly inside the element and a three node element is chosen. For these linear triangular toroidal elements, interpolation functions for C^°^ problems are the natural or area coodinates function . The L. 's are defined as L. ( P, C) 2 A (ai + bi p + c ) (7.6) where a . i = PjSc " Pk?j : ci " Pk - pj and bi = ?j " ?k (7-7) where (i, j, k) have module 3 and permute cyclically, ( Pj , C ^ are the coordinates of the nodal points (vertices) of a triangle numbered counter-clockwise 2 A (7.8) A is the area of the triangle The temperature within this element is expressed by the equation 3 (e) (e) [L1 L2 L3] <! 02 (e) (7.9) 51 where the 3^'s are the nodal values of the temperature In element (e) in the local numeration. Substituting 3^ 6 ^ given by Equation (7.9) into Equation (7.8) and considering that the integral should be performed for i from 1 to 3 we may write Equation (7.8) in matrix form as UKX] + [K ] ) <3>(E) = {K ) 3x3 3x3 1x3 1x3 (a) where ij 3L. 9L 3L 3L. . { - 1 + ( 1 + 2VL.) 1) dD16' (b) (e) 3P 3P 3C 1 3C (7.10) Hij ° (e) L.L . dS i J (e) (c) K = rh(z) B (z) A. . o a ij L . dS (e) 1 (e) (d) The evaluation of [K„] , [K„] and [K.] is given in Appendix III. In A [KT] is calculated exactly, [K H] and lKA3 are calculated approximately for three different peripherial elements according to their orientation on the external surface with respect to the axis of the crystal ; i) vertical, ii) tilted and iii) horizontal peripherial elements. Trp*e) TTVb. , . K = (b.b. + c.c.) + 1 [3 p(e) + p.] Tij 2A 1 J 1 > 6 J 52 where i , j = 1,2,3 p, * P2 * P3 (1) Vertical boundary [KH] = 2Trh(z)r()Pllj 1/3 1/6 0 1/6 1/3 0 0 0 0 <K } = Trh(z)r B (z)pl A 0 a 1 j 1 1 >• 0 (7.11) (i1) Tilted Boundary 27Th(z)r( hi 12 3P. + p. P. + P. P. + P. P. + 3 p. 0 0 0 TTh(z)r0 Ba(z) (pi + 2 p ) Al 1 1 0 53 (Hi) Horizontal boundary — (e) The same as in (i) but replacing pby pv ' 1^ is the length of the element at the boundary. Details of the numerical calculations including computer coding, type of mesh, mesh refinement, matrix assembly and matrix calculations are given later. 7.1.2 Stress Field Model To calculate the thermal stresses in the crystal the following assumptions are made : i. the body is linearly elastic and isotropic ii. the thermoelastic field is axisymmetric iii. the crystal is free from traction at the surface, i.e. at the surface o x n- = 0 , n is a vector perpendicular to the surface of the crystal. 7.1.2.1 Governing and Element Equations To obtain the element equation, the minimum potential energy principle was used. The strain energy for a linear elastic solid is defined as U (u,w) = 1 p 2 [e] {a} dv (7.12) V where (a) is the stress tensor and [e] is the strain tensor which for the present case have the following components 54 { o> TZ (7.13) The stress, according to Hook's Law and the Duhame1-Newman constitutive equation that accounts for the thermal strain, is io) [C] ({e} - {e0>) (7.14) where [EQ] is the strain corresponding to a free expansion of the solid due to temperature 'differences with respect to reference temperature TQ . This strain for an isotropic material with a coefficient of thermal expansion a is a( T V a (7.15) 1 0 [C] is the compliance mat r i x which for this case 1- V V V 0 E V 1-V V 0 [C] — V V 1- V 0 (1+V) (1-2V) 0 0 0 1 -2v 2 (7 . 16) with E the Young modulus and v the Poisson's ratio. The { e) strain can be derived from the displacement {d} as 55 ie ) (7.17) where [B] is the following matrix of derivatives [B] 3r J r 0 _3 3z~ "3~i~ 3 3r (7.18) Taking into account the relation among stress, strain and displacement, the potential energy can be written as II(u,w) = U (u,w) [d] [B] [C] [B] {d} 2[d] [B] [C] {eQ} dV (7.19) When the body is subdivided into small elements, it is assumed that within each element with g nodes the displacement field is approximated by <;>,e) = te!},e' = {i^O= [N],d, (e) (7 . 20) 56 where [N] are the interpolation functions matrix of order g and (e ) {d} are the element nodal displacements. In this case the minimization process may be carried out element by element. As a result the force-displacement equation for the element is obtained as 2gx2g 2gxl 2gxl [k](e) {d}(6) = {F}(6) {d} (e) {d> {d}, (e) (e) w. w. {d} (e) g w g (7.21) 2gx2g [k] (e) 2gxl (F) (e) [k] [k] [k] (e) 11 (e) il (e) gl {F> {F} { F } (e) 1 (e) 2 (e) g ™[V ^ [klj^ tk] tk]^^ [k] (e) In (e) ig (e) gg (7.22) 57 In the stiffness matrix [K], the submatrices [k]^j are the stiffness matrices at node i and have the form [k] (e) i j (e) 2x4m/ x 4x4. . 4x2. . [B]{(e) [C]<e> [B]<e> dV [B] (e) 3N 3r N 3N 3z 3N 3z 3N , 3r (7.23) The initial force matrix {FQ^ (e) l s [F }(C)  1 O' i (e) 2x4 . .4x4 4x1 . . [B]\(e) [C] fe0]{e, dV (7.24) 7.1.2.2 Requirements for the interpolation functions. Since only first order derivatives of displacement appear in the integrand for the potential energy, compatibility requires an interpolation function with continuity of displacement at the 58 element boundary only. However, for completeness, the Interpolation functions must be able to represent rigid body displacements and constant strain states within an element; that is uniform states of displacement and first derivatives of displacements respectively. This C^1^ continuity is satisfied with interpolation functions containing linear terms such as the natural coordinates used for the temperature field. This means that elements which are linear or of higher order are suitable. For convenience, before obtaining the specific form of element matrices, the node equation is written in terms of non-dimensional quantities as 2x2 . . 2x1 . 2x1 . . fk'lij {d'/j = <V>i (a) (7.25) where E {d.}(e) = {d}(e) (fe) a{ T - T ) r MP 0' l0 (e ) {k')!^ = / [B']j [C] [B']J6) PdPdC (c) where [B']<e) = rQ [B][e) (a) (1 + V) (1-2V) [C] = [C] (b) E 59 {F- ><e> ir O'l T(e) 6d / [B1 ]J [C')< i (7.26) V pd pd C ( c ) 7.1.2.3 Formulas for Stress Calculations The stress tensor is calculated at each node i in element e using the following equation ,(e) o. o. > -1- V V V V 1-V V v v 1-v o 0 1-2 V > - < 6.. 1 V (7.27) The strain components at each node are calculated from the nodal displacements as ie'}{e) - [B1]!6) <d')!e) (7.28) For nodes at the axis of the body eQ is substituted for e . 0 p The stress calculated in this way is non-dimensional. To obtain the stress in absolute units the following relation is employed 60 {0> (e) Ea (T MP (1-2V) T ) (7 . 29) 7.1.2.4 Linear Elements The linear elements are triangular toroids with nodes at the vertices of the triangle. The interpolation functions for these elements are the natural coordinates L , L0 and L and the displacement is written as {d} (e) L1U1 + L2U2 + L3U3 L1W1 + L2W2 + L3W3. (7.30) Making the corresponding calculations and substitutions, the terms of the stiffness and initial force matrices are [k,,ij} 11 12 21 22 (a) 11 (1-2 V) P v [(1-v) b.b. + I j c.c ] — + - [b.+b.] + 2 J 4A 6 J + (I-V)A ;-P dpd? (b) 12 ( 1 -2 v) [ vb . c . + i J ciV P v 4 A 6 + c . J (c) 61 21 (l-2v) p V [ vb c + c b ] + c (d) J 1 2 J 1 4A 6 22 [(l-V) C^Cj + (l-2v) blbj] (e) 4A (7.31 ) 2 A (bj p + (f) F ' (g) An Inspection of the above relations shows that all the terms can be evaluated exactly except the term containing 1/p which should be integrated numerically. One intermediate approximation consists of taking average values of p and (~p and L = 1/3) for that term only. Once the displacement field is calculated the nodal strain and stresses at each element can be derived by back substitution in the corresponding equations. The stresses at the centroid of the element are the average stresses. The stress associated with each node is the average value of stresses from all elements sharing the node. 7.1.2.5 Quadratic Element The quadratic element has six nodes, three nodes at the vertices and three nodes at mldsides. Interpolation functions are 62 built from the natural coordinates. A detailed calculation of stiffness and force matrix elements is given in Appendix V. It is important to note that the nodal [B] matrix of these elements includes only derivatives of the non-zero terms of displacement at that node instead of the derivative of all the terms in the interpolated displacement in the element. More specifically, in the element the displacement components are interpolated as usual as i \ 6 i \ 6 uie; = Z N.u. w*6' = £ N.w, (7.32) j=i 33 j=l J J where Uj and w.. are the nodal displacements and N the interpolation functions. For instance, by definition , . 3u(e) 6 3N. ele; = = Z J— u (7.33) 3r j=l 9r 3 and a true quadratic element will have a B element at node i which is 6 3N. B1 1 ( £ — N.) (7.34) i j=l 3r J N, = 0 NJ.- 1 J * 1 However in the pseudo-quadratic elements employed here, only the term N^. u^ is considered, i.e. 3N B = Ul (7.35) i or 63 It is also important to note that for the calculation of stresses the full [B] matrix is used in order to obtain representative and consistent values of stresses. 7.1.3 Von Mises and Resolved Shear Stresses The stress tensor in cylindrical coordinates obtained with the finite element method gives a quantitative description of the thermally induced stresses. For a dislocation generation analysis, these stress components should be transformed into quantities that can be associated with the plastic deformation of the material. This stress field can be described in terms of the Von Mises and Resolved Shear Stresses (RSS) and these values must be compared to the yield stress and Critical Resolved Shear Stress for GaAs to determine if dislocations are generated. The Von Mises Stress is related to the principal stresses by GVM = «J 1/2 [<0l ' 02)2 + (a1 ~ a3)2 + (o2 - o3)2] (7.36) The principal stresses, by definition, are the values of the stresses that diagonalize the stress tensor. The RSS1s are obtained by resolving the local stress tensor on the slip planes and slip directions. In GaAs the glide system is formed by the {111} planes and <110> directions giving 12 slip combinations. The procedure as well as the detailed calculation to obtain the twelve RSS components is given in Appendix VI. 64 The twelve RSS for a crystal grown in the [001] direction are given in Table 7.2 It can be seen that independent of the stress level, at every point in space there are three combinations of planes and directions that have the same resolved shear stress, these are (111) [110] (Til) [HO] (Til) [110] This leaves only ten different stress levels which will be called "glide modes". The difference between modes labelled a and b is the sign of the term involving the shear stress. The ten modes are listed in Table 7.1 Three dimensional stress contours are difficult to produce, therefore for the analysis of stress distribution two sections of the crystal are used, sections normal and parallel to the axis or growth direction. For normal sections stresses as given in Table 7.1 are used. For parallel sections the (010) plane is chosen in view of the experimental evidence that higher dislocation densities occur in that plane. For these planes 8 = 0. Substituting in the given equations for the ten modes gives the results shown in Table 7.2 It can be seen that in this plane there are only five different stress levels with the corresponding multiplicity. The RSS are compared with different critical stresses given in the literature for yield, dislocation generation and 65 multiplication for doped and undoped GaAs as described in section 8.1. Table 7.1 Resolved Shear Stress Components in (001) Crystals Plane|Direction| Resolved Shear Stress Node [110] /6 {-(a -Oq)COS26 + /2T JCOS ( 8 + TT/4 ) } 6 P y per I a (HI) [Oil] /6 {-(0 -o0)/2sinesin(6 + Tr/4) + (a_-o0) + ~g" P o t, a + ap£.cos6) II a [101] /6 {o -a J/2cos9sin( 9+TT/4 )-( o -oQ) -P e s 6' -Tp?sin6} III a [101] /6 {(a -0Q)/2cos6cos(e + 7r/4)-(ar-oa) ~~g P 0 t, t) -TpCsin9} IV a [HI] [Oil] /6 {(o -oQ)/2sin6cos ( 9+7T/4 ) + ( o -oj g P 0 CO -T _cos6} PC V b [110] /6 (- (ap-oQ) cos26 + /2Tp^cos (6 + 7T/4 ) } I a [Oil] j/6 {-(a -Oo)/2sin9sin(e + Tr/4) + (or-afi) -6 P e c e -T ? COS0} II b (111) [101] /6 ((a -o„)/2cosesin(9+77/4)-(a^-o.) + -g- p 8 C 6 +Tp?sin9} III a [110] /6 {-(o -ort)cos26+/2T COS(9+TT/4)} g P 6 p? I a Cont./ 66 Cont./ [Oil] /6 6 {(a -oQ)/2sin6co8( 8+TT/4) + ( o_-oQ) + p 0 t, 0 + T rcos 6} p£ V a (111) [ToT] j/6 6 {(a - oJ /2cos 6cos ( 6+ n/4 ) - ( a - o.) + P e c e + T .sine) p£ IV b [110] 6 {-( 0 - 0Q)C0S2 0-/2 T ,,008(9+^/4)} p o pt, I b 67 Table 7 . 2 Resolved Shear a [001] Crystal Stress Components in a (010) Plane for Node | RSS | Previous Modes | Plane | Direction | 1 A 6 I a (111) [110] (111) [HO] (111) [110] 2 /6 6 I b (111) [110] 3 6 11 V a a (111) [oil] (111) [011] 4 jM 6 [( O - OJ-T ] 11 V b b (111) [Oil] (111) [Oil] 5 6 [ o - o J P K 111 III IV IV a b a b (111) [101] (TTi) [ToT] (Tii) [ToT] (ill) [10T] 7.2 Modelling for Cooling After growth Once the crystal has reached its final length, it is pulled up to separate it from the melt and initiate cooling to room temperature. The effect of cooling in generating dislocations has not been established in the literature. However, the cooling 68 practice may induce thermal stresses that will generate more dislocations, particularly in the hot tail end region. In order to study the effects of cooling in generating dislocations the following mathematical approach has been developed. To obtain the time dependent temperature field an analytical solution was chosen. The selection of this method was based on two factors. a) Solutions are easy to obtain and numerical evaluations are cheap compared to numerical methods, b) The importance of cooling has not been sufficiently established to justify the use of expensive and time consuming numerical methods. The analytical temperature fields were calculated making the following assumptions : 1. The temperature field is time dependent and axisymmetric. 2. The crystal is a finite cylinder with constant cross-section. 3. The crystal is immersed in a medium (gas or liquid) at a constant temperature. 4. Newton's Law of Cooling applies at the crystal surface ; the heat transfer coefficient is constant and its value is determined by the temperature and nature of the surrounding medium. 5. The initial temperature of the crystal is axially dependent (radial temperature gradients during growth are 69 small) ; and it is given by a parabolic function that fits the temperature field at the axis of the crystal at the moment it is pulled from the melt. With these assumptions, the temperature fields satisfies the following partial differential equation and conditions using non-dimensional quantities as before : 3 33 (P ) + 3p 3p 323 33 3t (7.37) where t (K/r0*) t Initial Condition at t 60 = f(P) P0 + Pj S + P2 C (7.38) Boundary Conditions, with h1 = hr at t > 0 0 33 3p 33 3C 33 3C -h' 3 P =1 •h' 3 h1 3 5 =o (a) (b) (c) (7.39) 70 The solution to this problem as shown in Appendix V is given by 2 * 3(p.£.t*) = 2h'[Z CAX) e~X 1 J.(Ap)] X 1 0 2 * {£ C2<Y) eY 1 [YcosyC + h'sln Y q (7.40) Y where the \- values and y - values are the solutions of the following algebraic equations AJjU) = h' J0(A) (a) 2y h' (7.41) tanYCt = — -j- (b) y - h1 with Cj <M = ~2 ~2 (a) (h'd + A ) J0(A) and 11 C2 (Y) - (b) II = siny^ {pQ + Pj (Ct + h'/Y2) + P2(C2 - 2/Y2) 2 COSy h + 2 Ct h'/Y } + <-P0h' + pl(1 " h'Ct) 2 2 P0h' P1 2p?n' + P2(2?t " h'Ct + 2n'/Y2} + -£— - -1 - —f- (c) Y Y Y 71 €t ,2 L,2, ( Y + h' ) + h (7.42) (d) The temperature field calculated at a given time is employed to calculate the thermally induced stress field, with the aid of the finite element method described in section 7.1 . From these stresses the Von Mises and RSS are derived and compared to temperature dependent critical values for yield or dislocation generat i on. 7.3 Analytical Solutions Analytical solutions for the temperature and stress fields were obtained and numerical evaluations of these solutions were compared with those obtained with finite element methods. The analytical solutions developed were obtained for simplified boundary conditions. 7.3.1 Analytical QSS Temperature Field The quasi - steady state differential equation (7.1) is solved analytically for a simplified problem in which the following conditions are adopted : 1. Planar solid - liquid interface 2. Cylindrical crystal with constant cross-section 72 3. Uniform heat transfer coefficient h and constant ambient temperature Tfl 4. Constant temperature at the ends of the crystal From these, the boundary conditions are written as 93 " 9p p=l h1 3s (a) 31 = 1 (b) (7.43) <5=0 31 =0 ; E = ^- (c) 0 where is the total length of the crystal. The solution to the heat conduction equation (7.1), subject to the specified boundary conditions, is given by the following equation in term of the Bessel function of zero and first order JQ and Jj -kxC -2kxCT kxC v£ J0( ^P) te - e e g(p. O = 2hr ev^ Z ; ] (7.44) A ( A +h r/)Jn(A) -2k, c_ 0 0 (1 - e A M where the values are the solutions of the algebraic equation A Jj(A) = hr0JQ(A) (7.45) and k2 . v2 + x2 A Details of the calcuations are given in Appendix VII 73 (7.46) 7.3.2 Analytical solutions for the Stress Field Analytical solutions for the stress field were obtained using two different approximations : plane strain and axisymmetric. Two types of temperature fields were employed, temperature fields that depend on radius only (no axial temperature gradient) and general axisymmetric temperature fields. 7.3.2.1 Plane Strain Approximation In this approximation it is assumed that the displacement field consists of primarily a radial component, with the axial component constrained to maintain zero axial traction at the two ends of a cylinder. For the two dimensional axisymmetric temperature field calculated in section 7.3.1, the three components of non-dimensional stress from Appendix IX are 2hr hr Jj(Xp) 1 ] (a) o P Xp J0(A) 2hr ( Jj(X P) - J0(XP))] (b) o e Xp o 2hr (c) 74 where (7.47) "k^ -2kA *t kA 5 e - e e Kx (C) = (d) (1 - e A l)(\2 + h2rQ2) For a radial temperature field given by 3 = -P2 (7.48) the stresses are given by the following relationships 0p = 1/4 (p2 - 1) (a) OQ = 1/4 (3p2 - 1) (b) (7.49) 0? = p2 - 1/2 (c) Details of the calculations are given in Appendix VII. 7.3.2.2 Axisymmetric Solutions Analytical solutions using the axisymmetric approximation were obtained using Love's and Goodier's potentials. Goodier's potential gives a particular solution for the thermoelastic stresses while Love's potential gives a general solution for the isothermal elastic stresses. A combination of both potentials is necessary to satisfy the boundary conditions. In this case the body is a cylinder with constant cross-section and finite length. Solutions are obtained using finite Fourier transforms for 75 axisymmetric and radial temperature fields as in the plain strain case . Details of the calculations and solutions are given in Appendix IX. For purposes of comparison with the finite element method, non-dimensional stress component were obtained by multiplying the non-dimensional stresses by, a) 1 - 2v for the axisymmetric approximation and b) (1 - 2v) / (1 - v ) for the plane strain approximation with v = 0.29 76 CHAPTER 8 EVALUATION OF THE MODELS In this chapter the results of the models are compared with the analytical solutions presented in the last chapter. In addition the temperature fields obtained with the finite element method are compared with temperature measurements in a high pressure Melbourn puller during growth. The model is evaluated on the basis of these results. 8.1 Programming and Input Parameters Based on the numerical scheme presented in the last chapter, programs in FORTRAN language were written for the calculation of temperatures and stresses in the GaAs crystal during growth. Flow charts are shown in Figures 8.1 and 8.2 The flow charts corresponding to the computer programs employed for calculating the stresses with linear and quadratic elements are essentially the same. The mesh was automatically generated using a computer program with a flow chart as shown in Figure 8.3. Several different meshes using triangular elements were tested in order to select the one which gave the best results. A Choleski method was selected for solving the linear system of equations that resulted from the finite element method. In the case of the temperature field calculations, the test employed for the selection of the solution method and optimum mesh 77 ( START thermal dlffusivity temperature profile boron oxide thickness pressure growth velocity crystal geometry : rQ, z , cone angle, seed length, seed length, seed diameter. •esh data : number of nodes, elements, nodes with constant temperature (interface), boundary segments (vertical, tilted, horizontal) nodal coordinates, system topology. INITIALIZE : [KT], [KH], [KA], 0 For each element EVALUATE : [kT] e ASSEMBLE global matrix [] without regarding boundary conditions 78 For each boundary element ACCOUNT FOR convection - CALCULATE corresponding heat transfer coefficient and ambient temperature - EVALUATE [KH] and [KA] - ADD [KT] and [KH] For each node with specified MODIFY [KA] and [K^] temperature SOLVE the linear system [K ] {g} = [K ] using Choleski's method. A PRINT temperatures to be read by PFEMS ^ PLOT temperature contours STOP Figure 8.1 Flow chart of the computer program to calculate the temperature field in the crystal using a finite element method. ( START ") INPUT Poisson ratio crystal dimensions ; r , Z •esh data : number of nodes, elements, nodes with specified displacement, nodals coordinates, system topology nodal temperatures INITIALIZE : [K'] , {F^} For each element, at each node (e) EVALUATE FORM [k1 ] <F' } (e) ij * 'i using quintic order integration matrices [k']e and (F^}e in element ASSEMBLE Global Matrices [K'] and {F^} without regarding nodes with specified displacement 5 80 2 ACCOUNT for nodes with specified displacement SOLVE OBTAIN the linear system [K1] {d1} using Choleski's method and displacements u, w <F0} For each element at each node : CALCULATE strain components e , e„, e , Y P e c pc CALCULATE stress components o , o„, o , T P e C PC . At each node CALCULATE average stress calculated from all elements sharing the node CALCULATE principal stresses o , 0 , o J. O CALCULATE Von Mises stress 0 V Fl 81 PRINT Output Stress components and Von Mises stress at all nodes PLOT Von Mises stress contours STOP Figure 8 . 2 Flow chart of the computer program for calculations using finite linear and elements. the stress quadratic 82 ( START ) /^INPUT : element size, crystal length, seed width, seed length, cone angle INITIALIZE : last node and last element calculated LAST, LASTEL. CALCULATE : number of nodes in radial and axial directions recalculate element size for the bulk crystal GENERATE first row of nodes at the interface For each axial position in the bulk crystal, GENERATE nodes in the radial direction 83 2 For each eleient in the bulk crystal, GENERATE bulk crystal topology CALCULATE number and dimension of blocks in the cone For each axial position in the cone, CALCULATE number of nodes in the radial direction CALCULATE element size In radial direction GENERATE nodes in radial directions For each element in the cone, GENERATE cone topology CALCULATE number of nodes in radial and axial directions In the seed 5 84 © For each axial position in the seed, GENERATE nodes in the radial direction For each element in the seed, GENERATE seed topology DETERMINE segments where convection occurs, - vertical segments - tilted segments - horizontal segments /PRINT Data to be read by PFEMT and PFEMS / STOP Figure 8.3 Flow chart of the mesh generator computer program. 85 consisted of applying uniform boundary conditions on the surface of the crystal. These conditions were unit ambient temperature in the medium surrounding the crystal, constant heat transfer coefficient and unit temperature at the solid liquid interface. The temperature field in the crystal in such conditions should be uniform and have unit value. The combination of mesh type and solution method selected gave temperature fields for a coarse mesh, which were 99 % of the theoretical value. In the case of the thermoelastic stress calculations with the FEM, application of a unit temperature field throughout the crystal gave nondimensional stress values less that 10-4. in such conditions the theoretical stress should be zero. In addition, manual tests were carried out using simple meshes to ensure that the stiffness element matrices were symmetric and positively defined (eigenvalues should be positive). To minimize the computer memory requirements, which is critical for large systems, the stiffness matrix for the whole system was assembled directly in a linear array. The output of the computational effort of the temperature calculations consisted of nodal temperatures and isotherm plots in the crystal. The output in the stress program consisted of nodal displacements, nodal stress components in cylindrical coordinates, nodal Von Mises stresses and Von Mises isostress contours in the crystal. 86 Two other programs were written for calculating the maximum component of resolved shear stress in planes perpendicular and parallel to the crystal axis. These planes correspond to a (001) plane and (010) plane respectively. Similar plots are obtained of the difference between the MRSS and experimental values of critical resolved shear stresses for dislocation generation (CRSS) or yield (YS). Values of CRSS for doped and Te-doped [CRSS(Te)] GaAs have been employed for the comparison. These values are temperature dependent. The flow charts of the programs are shown in Figures 8.4 and 8.5. In all cases the output also consists of nodal values of the corresponding stress and the mode (specific plane and direction) which gives the maximum component of stress. The computer programs for the four calculations are listed in Appendix XI. 8.1.1 Input Parameters The input parameters for the calculations in the different computer programs depend on the stage being modelled. For the mesh generation, only geometric parameters are given, they are : 1) crystal dimensions : - radius - length - cone angle - seed length - seed radius - interface shape 2) element size given by the distance between nodes. /'INPUT number of nodes stress components nodal coordinates temperatures At each node - CALCULATE five components of the resolved shear stresses In (010) planes (RSS) - OBTAIN maximum component of RSS (MRSS) » GET node - SUBSTRACT corresponding critical resolved shear stress (MRSS > - CRSS) PRINT : MRSS - CRSS, mode PLOT : MRSS - CRSS contours ^ STOP^ Figure 8.4 Flow chart of the computer program to calculate and plot RSS in a vertical (010) plane INPUT - stress components in radial direction at Z - temperatures GENERATE polar grid At each polar node - CALCULATE ten components of the resolved shear stress (RSS) - OBTAIN maximum component of stress (NRSS) ; GET mode - SUBSTRACT corresponding critical resolved shear stress (MRSS - CRSS) L PRINT : MRSS - CRSS, mode PLOT : MRSS - CRSS contours ( STOP ) Figure 8.4 Flow chart of the computer program to calculate and plot RSS in a (001) plane 89 The input for the temperature program is the nodal and element configuration, the nodes at the interface and the segments where heat is transferred to the surrounding atmosphere. The physical data includes gas pressure, boric oxide height and temperature profiles in both media. From these the heat transfer coefficient at the corresponding temperature is calculated in the program. The heat transfer coefficient relative to the thermal conductivity includes heat transfer by radiation and convection h = h + h r c The heat transfer coefficient for radiation was derived by 19 0 191 Jordan ' from the Stefan-Bo1tzman equation leading to the following equation : h = (2 . 27 x 1011/ K ) £ T 3 r t a where e is the total emittance of GaAs and is a function of temperature and doping level. The convective heat transfer coefficient to the gas and B^Og layer was calculated assuming heat is transferred by free 191 convection from a vertical wall into a fluid and is given by T - T 1/4 h rem"1] = ( -) h" p1/2 C 1 where .548 _JL_ [p2 G ^ ^ /K^J1/2 K. a 90 h rod tooo 1100 1200 1300 1400 TEMPERATURE ( K) Figure 8.6 Estimated radiative and convection heat transfer coefficients for GaAs/B 0^ (1), He (g), Ng (g) and A (g) as a function or ambient temperature. The numerical labels are the product of the carrier concentration X crystal diameter in units of cm Figure 8.7 Total heat transfer coefficient h, in B2°3 an(* ar£on as a function of the ambient temperature T (1) Total heat transfer coefficient in B2°3' Total heat transfer coefficient in argon pressurized at 30 atm. 91 K = ambient thermal diffusivity a p = ambient density a ag = coefficient of thermal volume expansion Cp = heat capacity u = viscosity a p = pressure 1 = height of the fluid column for a liquid medium p is taken as p = 1. The numerical evaluation of the heat transfer coefficients 191 was done by Jordan et al. using temperature dependent values of the physical quantities involved. The results for a gas pressure of 1 atmosphere are shown in Figure 8.6 for the different media. The total heat transfer coefficients for boron oxide and argon at 30 atmospheres are shown in Figure 8.7 In the program the heat transfer coefficients are evaluated by functions which fit the data and include the effect of temperature and pressure. Other physical parameters are the thermal diffusivity which is assumed to be temperature independent, and growth velocity. The same nodal configuration employed in the calculation of temperatures must be used for the thermal stress calculations. The nodal temperatures are the input for this program. In addition the Poisson ratio and Young modulus are given. The values of the most important parameters are listed in Table 8.1. 92 Table 8.1 Values of physical parameters used In the calculations Melting point Thermal conductivity Thermal diffusivity Thermal expansion coefficient Young modulus Poisson ratio a E / (l-2v) 1238°C 0.08 Watts/cm K 0.04, ± 0.01 (800 C - 1238 C) (cm /sec) 1.0 X 10~5 (°K *) 1 . 2 X 1011 Pa 0 . 29 2 . 86 ± 0.1 (800 C MPa °K_1 1238 C) The yield and critical stresses employed for comparison with the MRSS were measured by Swaminathan and Copley 211 and 16 3 Mildviskii et al. respectively. The yield stress measured at temperatures well below the melting point are extrapolated to this point ; an inverse exponential temperature dependence was assumed as the experimental results suggest. The CRSS for undoped 10 07 o.) as 1 i 1 1 .L /2V /too 1001 KO t00 It Figure 8.8 Temperature dependence of the critical stress for dislocation generatjgn ig GaAs (1) Te-doped material, n_= 2 10 "187 X4°" 10 cm cm ; (2) Te-doped material, cm ; (3) Zn-doped material, p = 9 X (4) undoped material. 93 and doped GaAs close to the melting point are shown in Figure 8.8. It can be noted that the CRSS for doped material is about one order of magnitude larger than for undoped material. 8.1.2 Numerical Evaluation of the Analytical Solutions For the numerical evaluation of the analytical solutions for the different fields and conditions, programs in BASIC language were written. Programs were run in a VIC-20 microcomputer with expanded memory to 8 KRam. The A value, solutions of Equation (7.45) were obtained using graphic methods . The first five roots were used. The Bessel function JQ, , IQ and 1^ were 213 evaluated using polynomial interpolations for the functions The summation of the Fourier series was carried out using between 20 to 40 terms. This was necessary because of the typical slow convergency of this series. 8.2 Comparison of Model Predictions with Analytical  Solutions 8.2.1 Temperature Field Numerical solutions obtained using the finite element method were compared with analytical solutions for the problem described in Section 7.3. Calculations were performed for two values of the heat transfer coefficient h. The values selected, 0.3 cm 1 and 0.6 cm *, are in the range applicable to a crystal being grown in a pressurized argon atmosphere of 3.04MPa. The crystal radius was 20 mm and the height 40 mm. The results are shown in Figures 8.9 94 Figure 8.9 A comparison of finite element and analytical calculated temperature curves for h = 0.3 cm 95 F igure 8.10 and = 0.6 analytical cm 96 Figure 8.11 Mesh employed in the calculations of the temperature fields shown in Fig. 8.9 and 8.10. Number of nodes = 45. Number of elements = 64. 97 and 8.10 from which It is evident that the agreement is good. The finite element method gives temperatures which are slightly lower than the temperatures calculated analytically. A total of 64 elements were employed in the calculation and the corresponding number of nodes was 45. The mesh used is shown in Figure 8.11. Further refinement of the mesh did not appreciably change the results. 8.2.2 Stress Fields Finite element solutions were obtained for a cylindrical crystal for radial and axisymmetric temperature fields using linear elements. A crystal of radius 20 mm, and length 40 mm was discretized into a mesh of 256 elements which consists of 153 nodes. The stresses were calculated on the basis of a) the computed nodal temperatures and b) a uniform temperature in the element derived by averaging the nodal temperatures. For radial temperature fields, the radial stresses calculated by the finite element method for nodal and average element temperatures are plotted as a function of r/r^ in Figure 8.12. For comparison the corresponding stresses determined analytically are also shown for both plane strain and axisymmetric displacements. The radial stresses effectively coincide for the two analytical solutions and the finite element solutions using the average element temperature. The results for the finite element method using computed nodal temperatures is well below the other three. This large difference is attributed 98 CO I Q_ if UJ -02 -0 4 --06 --0 8 -r/rr Figure 8.12 Calculated radial stresses as a radial temperature fields (1) averaged element (2) Finite temperatures (3) Analytical Analytical-axisymmetric. function of r/r for Finite element with element with nodal plane strain (4) 99 Figure 8.13 Calculated azimuthal stresses as a function of r/r for radial temperature fields (1) Finite elemen? with averaged element temperatures (2) Finite element with nodal temperatures (3) Analytical-plane strain (4) Analytical-axisymmetric. 100 i 1 1 r -0-6 0 02 04 06 08 r/r. 10 Figure 8.14 Calculated axial stresses as a function of r/r for radial temperature fields (1) Finite element with averaged element temperatures (2) Finite element with nodal temperatures (3) Ana 1ytical-p1ane strain (4) Analytical-axisymraetric. 101 to the Incompatibility in the approximation of initial strain in comparison with the strain field in the linear element. If the nodal temperatures are used, without averaging, both the temperature and initial strain vary linearly within the element while the linear displacement formulation gives a constant strain within the element, which is inconsistent. When the nodal temperatures are averaged and assumed constant within the element, the initial strain is also constant and is consistent with the strain associated with the displacement field. The azimuthal and axial stresses, as a function of r/rQ for the same solution are shown in Figures 8.13 and 8.14 respectively. The results in Figure 8.13 are generally similar to the radial stresses in Figure 8.10 with the finite element nodal temperature method of introducing thermal strain giving a curve falling well below the other three. The plane strain analytical solution deviates from the other two a small amount with increasing r/r^. The axial stress curves in Figure 8.14 have a different configuration than the radial and azimuthal solutions. The best agreement is between the finite element average temperature curve and the axisymmetric curve as before. The finite element nodal temperature curve is not valid for the reasons given previously. The analytical plane strain values deviates from the other two giving maximum tensile and compressive stresses. To test the convergence of the finite element procedure, stress calculations were made with refined meshes for a radial Figure 8.15 Four steps in the mesh refinement used to analyse the conver gency of the finite element stress calculations. (a) NN = 15 NE = 16 (b) NN = 45, NE = 16 (c) NN = 45, NE = 64. (d) NN 153, NE = 256 (b) Quadratic elements. (a), (c) and (d linear elements. ,_, o to 103 0 -02 -03 -0-4 CM I •o. -0-5 2 -0-6 -07 -0-8 -0-9' NODES ELEMENTS L 45 64 • 1 Q 45 16 A L 153 256 L 15 16 • 2 L 45 64 A Q 45 16 O L 153 256 10 Figure 8.16 Calculated radial stresses as a function of r/r for different numbers of nodes and sizes of element (1) Finite element with averaged element temperature (2) Finite element with nodal temperatures. L = Linear element, Q = Quadratic element. 104 temperature field. Calculations were made as a function of r/r^ for a cylinder of half a radius in length. The different steps in the mesh refinement are shown in Figure 8.15. The results of the stress calculations are shown in Figure 8.16 using individual node temperatures and average element temperatures. The results indicate that large changes in the number of nodes and elements have no significant effect on the calculated stresses in both cases. The results also show that the large difference between the nodal and averaged element temperature is not associated with element size but with the incompatibility between strains as described above. It is also evident that the quadratic element formulation does not lead to a significant difference in the computed stresses. For the same number of nodes fewer and larger elements are involved in the quadratic element formulation, as seen in Figure 8.16, and hence the improved accuracy of the quadratic interpolation of displacement is compromised by the coarser grid. The finite element and analytical radial stress fields for an axisymmetric thermal field are shown in Figure 8.17. The temperature field corresponds to an h value of 0.3 cm * for a crystal of radius 20 mm and length 40 mm. The number of nodes and elements were 256 and 153 respectively. The finite element average temperature curve nearly coincides with the analytical axisymmetric curve for r/rQ above 0.5, deviating below this value. As before, the finite element nodal temperature curve differs markedly from the others. The analytical plane strain curve has a higher compressive stress than the others. 105 Figure 8.17 Calculated radial stresses as a function of r/r for axisymmetric thermal fields (1) Finite element with averaged element temperatures (2) Finite element with nodal temperatures (3) Analytical-plane strain (4) Analytical-axisymmetric. 106 The difference between the analytical axisymmetric curve and finite element average temperature curve is attributed to the slow convergence of the Fourier series combined with the use of the first order Bessel function in the analytical solution of the stress field which results in less precise values. Convergence in this series is oscillatory and values were calculated by averaging the last two terms in summations involving about 40 terms. The argument increases linearly with the number of terms and the 1^ Bessel functions diverge very rapidly with these arguments. This effect is not serious when radial temperature fields are used because of the simplicity of the function which is expanded (the unit function), as opposed to a complex z-dependent function which is expanded in the case of an axisymmetric two dimensional temperature field. The inclusion of the I functions was done in order to obtain a more general solution to the isothermal displacement differential equation. Recently solutions for similar equations have been obtained in semi-infinite cylinders including only the J Bessel functions in 16 8 a less general but better behaved solution The results in Figure 8.17 also show that for axisymmetric temperature fields the plane strain approximation gives stresses which are more than double the stresses obtained with the finite element method using the axisymmetric approximations. Larger differences were usually observed close to the ends of longer crystal where conditions deviate markedly from plane strain. This result shows that the plane strain approximation is not always good for the analysis of thermally induced stresses. It should be 107 noted that this result has not been reported in the literature214"217. In summary, it has been shown that there is good agreement between the finite element analysis using average element temperatures and the analytical solution with the axisymmetric assumption. The analytical solution with the plane strain assumption does not fit as well due to the restriction of axial movement in this case. It may thus be concluded that the linear finite element analysis gives stresses that are comparable to the axisymmetric approximation and with the plane strain approximation when that condition applies. 8.3 Comparison of Model Predictions with Temperature  Measurements The temperature distribution in a growing crystal calculated from the model is compared to the results of temperature 218 measurements made in a LEC GaAs Melbourn crystal grower. The temperature measurements were made with the array of four thermocouples shown in Figure 8.18. The thermocouples were attached to the seed holder and moved up with the seed as the crystal was grown. Temperatures were recorded for the entire growth period of the crystal, giving the results shown in Figure 8.19. To calculate the temperature distribution with the model, 2 the thermal diffusivity was taken as 0.04 cm /s. The heat transfer coefficients between the crystal and the argon and BO were determined from the values given in section 8.1. 108 Figure 8.18 Position of thermocouples in GaAs crystal. B = Boric oxide layer, argon pressure 3.04 MPa. Reference 218. 109 The ambient temperature distribution along the crystal, shown in Figure 8.19 (curve 4), is based on temperature measurements adjacent to a fixed point on the crystal at different times during growth, converting time to distance with the growth velocity. It is assumed that this temperature distribution is time independent. On the basis of this assumption, the measured ambient temperature is used to determine the heat transfer coefficient at each segment along the surface of the growing crystal. The temperature distribution within the crystal will change with time as the crystal is pulled from the melt. The calculated and measured temperatures along the crystal axis at four different crystal lengths are shown in Figure 8.20. A comparison of the calculated and measured temperatures is only valid at the point where the calculated temperature coincides with the measured temperature, at points marked A, B, C and D in Figure 8.20 for the four crystal lengths. Agreement between the calculated and measured temperatures at these points is close ; within 15°C. For the short crystal (55.0 mm), the measured and calculated curves effectively coincide, indicating that the measured curve is time independent. For the longer crystals the measured curves are below the calculated curve indicating temperature dependence, i.e. that the temperature at a given point decreases after being measured by the thermocouple as the crystal grows. The difference increases with increasing crystal length. 110 •20 0 20 4 0 60 80 100 120 140 160 Position Relative To Interface (mm) Figure 8.19 Temperatures measured with thermocouples 2, 3 and 4 in Figure 8.18 as a function of the relative position of the thermocouples with the interface. Reference 218. Ill ——i 1 1 1 1 1 1 1 1 1 r 1300 Position Relative To Interface (mm) Figure 8.20 Measured and calculated temperatures crystal axis at four crystal lengths. along the 112 Figure 8.21 Measured and calculated te outside surface of the lengths. The measured ambi shown. mperatures adjacent to the crystal at four crystal ent temperature is also 113 81 1 1 1 1 1 1 1 1 1 1 r l 1 1 1 1 1 i i • • I L 0 20 40 60 80 100 Position Relative To Interface (mm) Figure 8.22 Measured and calculated axial temperature gradients along the crystal axis at four crystal lengths. 114 The temperature distribution near the edge of the crystal, both calculated and measured, for four crystal lengths is shown in Figure 8.21. The ambient temperature distribution is also included in the figure. Note that adjacent to the interface, curve (a) was measured and curve (b) was used as the ambient temperature in the model since the interface is assumed flat and at the freezing temperature. Comparing the calculated and measured values at points A, B, C and D shows there is reasonable agreement between the two values. The calculated values are lower than the measured values by less than 25°C. As with the axial temperatures the measured values are below the calculated values over most of the crystal, the difference increasing with increasing crystal length. The axial gradient was also measured during growth by thermocouples 1 and 2 in Figure 8.18. The results are compared to the calculated gradients in Figure 8.22. The measured gradients are significantly below the calculated gradients near the interface. At these points the measured gradients may correspond partly to the melt and partly to the crystal. In addition, the presence of a convex interface, as is shown later in this work, gives lower axial gradients than a flat interface in this region. For longer crystals the measured gradients are about 10 % below the calculated values, the difference increasing with crystal length. At the points where the comparison is valid the measured values are above the calculated values but within the range of the scatter of experimental measurements. 115 In summary, the valid points of comparison of the calculated and measured temperatures are in good agreement. Accordingly, the temperatures predicted by the model are indicative of the real temperatures in the crystal. 8.4 The Temperature Model for Cooling - Programming and  Convergency To calculate the temperature fields based on the analytical time dependent solutions, a FORTRAN program was written. This program is shown in Appendix XI. The flow chart of this program is shown in Figure 8.23. The value of X were calculated using 219 cubic interpolation from tables , the first six roots were used. The Y-values were obtained from Equation (7.41b) using a Newton - Raphson method to find the roots. The first 30 values were used. With these six X values, the radial part of the temperature converged to 98 % of its final value. With 30 Y values the axial part of the temperature converged to 98 % of its final value. The non-dimensional temperature was therefore calculated as 96 * of the exact solution. A typical temperature field calculated during cooling is shown in Figure 8.24 a. The radius of the crystal is 27.5 mm and the length is 110.0 mm. Due to symmetry, only the right half of the crystal is shown. Close to the interface or tail end the axial temperature gradient changes from positive to negative. Also a high radial gradient is usually observed as opposed to what is observed during growth. In view of this and the fact that 116 ( START /INPUT - y eigenvalues - 3 eigenvalues - crystal radius and length - heat transfer coefficient, h - ambient temperature - initial axial temperature For each 3 : CALCULATE coefficient (3) For each Y : CALCULATE coefficient C (y) For each radial position : EVALUATE series in Bessel functions 5 117 1 For each axial position EVALUATE series in Fourier functions For each point in the grid CALCULATE temperature / / / PRINT nodal temperatures / j PLOT isotherms / STOP Figure 8.23 Flow chart of the computer program for the numerical evaluation of the analytical temperature fields during cooling of the crystal. 118 a be Figure 8.24 (a) Typical temperature field obtained during cooling units are 10 °C. (b) and (c) Von Mises stress contours (MPa) for the temperature field given in (a). (b) NN = 451, NE = 800. (c) NN = 1105, NE = 2048. 119 for the stress calculation average temperatures are used another mesh refinement was performed. This was done to determine if for a given mesh size the calculated stresses were representative. The results are shown in Figure 8.24 b and c. Figure 8.24 b shows the Von Mises stress field Induced by the temperature field in part (a) of the figure, calculated using a mesh containing 451 nodes and 800 elements. Figure 8.24 c shows the same stress field obtained employing a mesh of 1105 nodes and 2048 elements. It is seen that exactly the same stress pattern is obtained with stress values which are about 10 * less in the coarse mesh following an expected tendency. The computing effort involved, however increased quadratically with the number of nodes. Accordingly the coarser mesh was selected. 120 CHAPTER 9 RESULTS AND ANALYSIS The mathematical model presented in the previous chapters is used to study the effect of the different variables in generating dislocations during growth and cooling to ambient temperature. The variables related to crystal geometry during growth include the following 1. Cone angle (CA), given by the angle between the cone surface and the horizontal 2. Crystal length (CL), given by the distance between the solid-liquid interface and the start of the cone 3. Crystal radius (R), 4. Interface shape The variables associated with the growth process include the following : 5. Growth velocity (V) 6. Boron oxide thickness (B) 7. Thermal gradients (argon, AG and boron oxide, BG) 8. Gas pressure (GP) 9. Gas composition The variables associated with the cooling process include the following : 121 10. The nature of the media in which the crystal is immersed. This may be boron oxide or argon gas 11. Temperature of the cooling media 12. Thermal conditions in the crystal at the start of cooling. The effect of the above variables on the calculated stress fields and dislocation density are now considered. 9.1 Cone Angle Crystal and Growth Parameters Assumed R, 20 mm ; CL, 10 mm ; B, 10 mm ; BG , 100 °C/cm ; AP, 30 atm ; AG, 50°C/cm. The cone angles considered are 7.1, 30, 45, 54.7 and 65 degrees. The angle 54.7° corresponds to the cone surface coincident with the (111) plane. The calculated Von Mises stress distribution on a vertical plane for the five cone angles considered is shown in Figure 9.1(a-e). The maximum Von Mises Stress (MVMS) levels occur at the shoulder where the cone reaches the full radius of the crystal. High stress levels occur along both the cone surface and the crystal axis. The lowest Von Mises stresses (LVMS) are below the seed . 122 123 Figure 9.1 Von Mises Stress contours (MPa) in vertical planes for five cone angles. (a) 7.1°, (b) 30°, (c) 45 , (d) 54.7°, (e) 65°. Cone surface in (d) coincides with a (111) plane. Crystal radius, 20 mm ; crystal length, 10 mm ; B2°3 thickness, 10 mm ; B2°3 gradient, 100 C/cm ; argon pressure, 30 atm. ; argon gradient, 50 C/cm. 124 \Table 9.1 Effect of Cone Angle on Thermal and Stress Fields Thermal Gradients, °C/cm| 1% V™ rOue* 4 sl_ : MVMS I AMRSS 1 I AMRSS/ Cone Angle (°)| Radial Axial | MPa MPa MVMS 55 18.4 8 . 7 0 .47 50 6 . 7 3 . 1 0 . 46 54 5 . 2 2 . 5 0 . 48 56 1 . 1 2 . 6 2 . 4 55 5 . 0 2 . 3 0 . 45 7.1 6.7 30 . 4.1 45. 2.7 54.7 3.2 65 . 2.5 The largest MVMS, as shown in Table 9.1, occurs in the crystal with the sharpest cone angle of 7.1°. The MVMS drops appreciably when the cone angle is increased from 7.1° to 30°, and decreases further by a smaller amount between 30° and 45° . The MVMS reaches a minimum for a cone angle of 57.4° then increases between 52.4° and 65° . Similar behaviour is observed when the LVMS is considered as a function of cone angle. In order to account for the effect of the cone angle on the VMS two sources of thermal stress are considered namely the temperature fields and the geometrical constraints. The temperature field is the origin of thermal stresses. The stress level is roughly determined by the thermal gradients. Generally, large radial thermal gradients combined with large axial gradients produce large stresses in cylinders. If the axisymmetric body does not have constant cross-section, additional stresses may develop due to geometrical constraints. 125 For the crystal geometry considered here temperature gradients are calculated as follows. The axial gradients are calculated from the temperature differences between the interface and a point in the crystal axis 10 mm above, labelled A and B respectively in Figure 9.1(e) . The radial gradients are calculated from the temperature differences between point B and C which are also shown in Figure 9.1(e). The results are given in Table 9.1. From Table 9.1 the axial gradients are observed to be relatively insensitive to the cone angle, varying by about 3 * about the mean gradient of 53°c/cm. On the other hand the radial gradients vary appreciably about the mean gradient of 4.2°C/cm. The gradient drops sharply from 6.7°C/cm between 7.1° and 30°, drops further at 45°, rises at 54.7° and then drops again at 65°. Considering the MVMS values in Table 9.1, the drop in MVMS between 7.1° and 30° can be accounted for by the change in radial gradient. The minimum MVMS for 54.7° is attributed to the geometrical constraints arising from the variations in radius of the crystal at the cone, as well as the radial gradient. For a zero cone angle, conditions similar to plain stress will prevail at the cone surface, with zero axial and shear components of the axisymmetric stress tensor. Under such conditions the VMS will have maximum values because there are only two large non-zero diagonal stress components. For the same thermal conditions, when the cone is not zero a similar stress level will have a different distribution of stress components giving a third non-zero 126 principal stress. The difference between the principal stress components decreases as the cone angle increases, giving the observed VMS dependence with cone angle. For cone angles larger than 54.7° another factor should be included in the analysis. Under these conditions the crystals are effectively larger than the low cone angle crystals. Therefore the constraints are such that conditions closer to those of plane strain apply. In such cases it was shown in the last chapter that stresses are larger than under the more general axisymmetric conditions. From the above discussion, at low cone angles the VMS is expected to decrease with increasing cone angle. At high cone angles, however, the VMS is expected to increase with increasing cone angle. Therefore a minimum in the VMS should exist at intermediate cone angles. In this case that minimum is observed for a cone angle of 54.7°. The maximum component of the resolved shear stress (MRSS) in vertical (010) planes for the five cone angles considered is shown in Figure 9.2(a-e). The MRSS stress distributions are similar to the VMS distributions. The dependence of the MRSS on the cone angle is determined by considering the absolute maximum MRSS (AMRSS) developed in the crystal which is representative of the stress level in the whole crystal. The AMRSS occurs at the shoulder as is shown in the VMS distributions. The AMRSS for the angles considered is listed in Table 9.1 column 5. For low cone angles the AMRSS decreases rapidly similar to that shown for the VMS. For cone angles of 45° and larger the AMRSS remains 127 128 Figure 9.2 Maximum resolved shear stress (MRSS) contours in MPa, in vertical (010) planes for five cone angles, (a) 7.1°, (b) 30°, (c) 45°, (d) 54.7°, (e) 65°. Cone surface in (d) coincides with a (111) plane. Conditions are the same as in Figure 9.1 . 129 essentially constant. The large drop in the AMRSS at low cone angles supports the conclusion that the stresses are dependent entirely on the thermal field changes rather than on geometrical constraints. At large cone angles, the MRSS differs from the VMS. The difference can be seen in Table 9.1 column 6 where the AMRSS/MVMS is listed. For all cone angles except 54.7° the AMRSS is less than half the MVMS. At 54.7° a singularity occurs such that the AMRSS is more than double the MVMS. The singularity at 54:7° cone angle is attributed to the model assumption of no traction at the crystal surface and the difference in the definition of the VMS and RSS components. The condition of no traction requires the projection of the stress tensor on a direction normal to the crystal surface to be zero. In this case, for a given stress level, the stress components must accommodate in a direction parallel to the crystal surface. For a cone angle presenting a surface parallel to a (111) plane, the stress components lie on the (111) plane. In this case the RSS components are expected to be at maxima since by definition they are projections onto these planes in the <110> directions. The VMS, on the other hand, is an indication of the shear stress in isotropic materials and therefore does not include the crysta 11ographic properties of the material. These results show the advantage of using the MRSS over the VMS for the analysis of stresses generating dislocations. From the VMS results it may be mistakenly concluded that a 54.7° cone angle is beneficial in order to reduce dislocation generation. A 130 different conclusion can be obtained using the RSS which is the stress generating dislocations by glide. The results using the MRSS show that there is no such beneficial effect using a 54.7° cone angle. On the contrary, if growth conditions are such that stress levels given by the VMS are changed, the effect on dislocation densities is double at this angle. The present calculations differ significantly from the 13 7 calculations reported by Jordan et al. . In the present case the MRSS is used, whereas Jordan et al. use the total resolved shear stress (TRSS) which is the sum of the twelve RSS values. For the 45° cone angle the TRSS distribution shown in Figure 9.3 is similar to the MRSS distribution shown in Figure 9.2(c), but the stress levels of the TRSS are seven times larger than the MRSS and three times larger than the VMS levels. Adding the components to make the TRSS is inconsistent with the concept of a vectorial or tensorial quantity since the magnitude cannot be calculated by a linear addition of components. Moreover the TRSS can be indefinitely increased by considering an unlimited number of secondary slip systems which is also inconsistent. The effective stress associated with dislocation generation and movement can be determined by substracting from the MRSS the critical resolved shear stress for yield or the value reported for dislocation generation. This is done in Figure 9.4 in which (a) the yield stress is substracted from the MRSS [CRSS (yield)], (b) the CRSS for dislocation generation given by Milvidskii and Bochkareev150'163 is substracted [CRSS (MB)] and (c) the CRSS 131 Figure 9.3 Total resolved shear stress (TRSS) contours in MPa for the 45 cone angle crystal. Compare the large stress levels of the TRSS with the stress levels for same crystal shown in Figure 9.1(c) for the VMS and Figure 9.2(c) for the MRSS. / 4 Contours of the MRSS (MPa) in excess of : (a) CRSS (yield) ; (b) CRSS (MB) ; (c) CRSS (MBTe). Shaded regions indicate areas in which the MRSS is less than the CRSS. 133 for Te-doped crystals is substracted [CRSS (MBTe) ] . The shaded areas in the figures indicate where the effective stress is zero. It can be seen that the MRSS is less than the yield stress in most of the crystal except in the centre and near the surface. The largest stresses occur in a region below the shoulder at A. The MRSS is always greater than the CRSS (MB) as shown in (b). Doping with Te in (c) gives regions in the crystal where the MRSS does not exceed the CRSS (MBTe) comparable to (a) . The specific directions and planes associated with the maximum resolved shear stresses in the crystal are shown in Figure 9.5. Plots of stress distribution for other cone angles are given in Appendix XIII. These plots show similar patterns to that shown in Figure 9.4 with the exception of the crystal with a cone angle of 7.1° (Figure 9.6(a-c)). 9.1.1 Effect of Cone Angle on Stress Symmetry The effect of cone angle on the MRSS distribution on planes perpendicular to the growth direction is shown in Figure 9.7(a-e). The planes in (a-d) are 7.5 mm from the interface, and the plane in (e) is 8 mm from the interface. The vertical direction in the figures is [010] and the horizontal direction [100]. In Figure 9.7(a), for the smallest cone angle (7.1°), the stress distribution is complex with low stresses at the centre followed by a ring with higher stresses which exhibits four-fold symmetry. 134 O-O-O-O-O-D <J> O A A A Q o o A A A LJ) 6 O A A • O o o AO o A 6 o • • o • ' \ o o o o o o. I I I I I I V I o I o o o o oxo o o o o, I o 6 o  o V o o o o o o o o ooooooooo o o o o o o o o o o o o o o o o o\>. \ 6 o o o o o o o o o <p o o o o o o o o o o o o o oNo. o o o o o o o o oooo ooooo o o o o o o o o o o o o • o o. o o. (III (Til ( TT I O (Ml 0 (II. (Ill • \ o-o- o o O o o • • • 1 • c > o o o o • • • • o-o- o o o o o o o o o o o o • • o 1 o o • • • • • A o o o o • • • o 1 1 o 1 o 1 ? o A • • • • • o o o o o o o o I o 1 o 1 ? o A • • • • • A o o o o o o o o A • • • • • • A o o o o o o 1 o 1 o 1 o A • • • • • • • A o o 0 o o 1 o 1 ? o A • • • • • • • • A A o o o 1 o 1 o A • • • • • • • • • • A A A 1 • I 1 o- o--o--o--0--o--o -O- -0--o- 0-•0-0-o- o-•o-1 -o (III (Til (in ( TT I (Til (Ml M I 01 C I I 0 3 I I T 03 (T T 0] COT I 1 COM] tO I I] (Oil] II on rT on [ToT ] l I OT] Figure 9.5 Slip mode distribution in the (010) plane for the 45° cone angle crystal corresponding to the MRSS distribution shown in Figure 9.2(c). 135 Midway between the centre and the edge of the wafer there is a ring with minimum stresses which exhibits circular symmetry. At the edge of the wafer the stresses increase to a maximum value. The stress at the edge is five times larger than the stress at the centre. This complex pattern is consistent with what is observed in Figure 9.6(a) at 7.5 mm from the interface. The ring at the center with higher stresses is due to the irregular contours in the upper half of the crystal, at A in Figure 9.6(a). For higher cone angles the stress distribution is simpler. The crystals with 30° and 45° cones shown in Figure 9.7(b) and (c), exhibit almost circular symmetry with low stresses in the central area which includes half the total surface area. Outside this region the stress rapidly increases towards the edge of the wafer. For both cone angles the stresses at the edge are about five times larger than at the centre. For larger cone angles a region of higher stresses starts to build up at the central region, as shown in Figure 9.7(d) and (e), producing the characteristic W-shaped stress distribution across the diameter. For the largest cone angle (65°) the stresses at the centre are three times larger than the minimal stresses at the mid radius position. At the edge the stresses are about eight times larger than the minimum values. It is shown later that the qualitative characteristics of symmetry presented here are general. This is because the symmetry patterns of the stress fields are independent of the thermal gradients, providing 136 the temperature profile remains linear and the growth conditions (radius, boron oxide thickness, etc.) remain unchanged. The density and distribution of dislocations in the crystal is related to the local excess stress above the critical resolved shear stress (MRSS-CRSS). When the CRSS (Yield) is taken as the CRSS, the residual stress depends on the cone angle. For the smallest cone angle the MRSS is always larger than the CRSS (Yield) and the symmetry is very similar to the symmetry of the MRSS as shown in Figure 9.6(a). For larger cone angles the MRSS in excess of the CRSS (Yield) is greater than zero in the central and peripheral regions in the wafer. This is shown in Figure 9.8 for the 65° cone angle. Similar results are obtained for the other cone angles. The stress distributions on cross-sections 2.5 mm from the interface are shown for cone angles 7.1° (Figure 9.9(a-b) and 45° (Figure 9.9(c)). In Figure 9.9(a) the MRSS distribution exhibits a W shape across the diameter and has eight-fold symmetry in the centre. The MRSS-CRSS (Yield) (Figure 9.9(b)) in the central region is below zero so that no dislocations or symmetry patterns will be evident. As the cone angle increases the eight-fold symmetry in the centre is replaced by four-fold symmetry as shown in Figure 9.9(c) for a 45° cone angle. In this case the dislocation symmetry will be evident. It can also be observed in this figure that the centre has stress levels three times larger than the edge. The stress distribution along the radius depends on the direction. In the [110] directions the stress starts to 137 igure 9.6 Contours in MPa of the MRSS in excess of (a) CRSS (Yield) ; (b) CRSS (MB) and (c) CRSS (MBTe). In most of the crystal the MRSS is larger than the critical values. The bump shaped contours (A) give complex stress distributions in perpendicular cross-sections. 138 a 139 140 C 141 d gure 9.7 MRSS contours (MPa) in cross-sections perpendicular to the crystal axis for five cone angles. (a) 7.1°, (b) 30°, (c) 45°, (d) 54. 7°, (e) 65 . Section (a-d) are 7.5 mm from the interface and section (e) is 8.0 mm from the interface. The horizontal direction corresponds to the [100] direction and the vertical direction corresponds to the [010] direction. 142 Figure 9.8 Contours of the MRSS-CRSS (Yield) (MPa) for the 65° cone angle crystal. MRSS contours are shown in Figure 9.7(e). At high cone angles MRSS levels are larger than CRSS at the centre and outside part of the wafer. a 144 145 Figure 9.9 Stress contours (MPa) in cross-section 2.5 mm from the interface. (a) and (b) in a 7.1 cone angle crystal, (c) in a 45 cone angle crystal, (a) MRSS contours, (b) and (c) MRSS-CRSS (Yield). For the 7.1° cone angle there is eight-fold symmetry at the centre in (a) which is not seen in (b) because stress levels are less than CRSS. For the 45° cone angle there is four-fold symmetry. 146 drop closer to the center than in any other direction ; in addition the stress remains low even close to the edge. As a result larger areas of low stresses are present in the <110> directions. Similar conclusions were obtained with the higher cone angles. In all the cases considered there is one common characteristic present. At the edge of the wafer eight-fold symmetry is always present with minima in the <100> and <110> directions. This is more apparent in Figure 9.7(d), where the maxima are symmetrically distributed. The eight-fold symmetry at the edge is associated with an eight-fold distribution of the slip mode. This is shown in Figure 9.10. It is also observed that the local minima at the edge in the <100> and <110> directions are associated with a change in the slip mode. 9.1.2 Effect of Thermal Conditions In order to assess the effect of the thermal conditions on the stress distribution during growth, stress fields are determined in crystals having a 45° cone angle for different thermal gradients in the boron oxide and argon gas, as listed in Table 9.2 . The calculated temperature distribution in the growing crys tal for a range of thermal gradients is shown in Figure 9.11(a-d). The isotherms are observed to be very flat for low thermal gradients with a small curvature at the shoulder of the crystal. 147 o ( 1 " " I ) [1101 0 ( 1 1 i ) LOT 1 ] • ( T" " 1 ) 10 II D ft ( I 1 ) L 1 OT 3 • (Tin [ TOT ] * (iT I ) 11 of] B ( I 7 I ) CO I I 3 e ( 7 I I ) [ OT I 3 C 0 I 0 3 •(III) (III) (Til) ( TT I ) [10 13 [T I 03 [MOD t I T 03 ^ft * ft ft A * * A . * ft /° ° * * ft ^° ° ° • ft A A ft s a oaaaaDftAB## / D°DODA(S • ••••Oft ft ft ft ft ft o o a A • ••••••ft fta 99 ft ft ft ft o o o s ft • • •••••4*8999 9 o o o o o • ft A ft 9 ft O O o • • ft ft ft ft • • • ft ft a a \ a a a a a^ a a a a a a ° o • • " - \ #Q9aaBBaaa^ ft • • a a a a a a a 00 •••aaaaaaa -^•••••••ftftB9999 000 e«dOAft»Be \ 9000900ftft<* B 999Z99**9 9 \a 00000**** 9 9 e0 00O9oooaaaaa 9ft •••••OO OOO O999[|003 0BOOO 00090 > • 00999999 B • 9 9 99999 •»»9099 9^ ••99999^ * • 9 9 9 J # • • 9 • 0 . V a • 9 • • ft • • • • a • a ft A ft • • • • ft ft 9 • • • ft ft • A ft • 0 ft • ft A • • 0 9 0 9 9 9 0 # 9 9 9 9 9 9 ft ft ft ft ft m * ft ft ft ft 9 ft ft ft 9 9 ft ft ft 9 9 ft ft ft 9 9 ft ft ft 9 9 * • 0 Figure 9.10 Slip mode distribution in a (001) plane corresponding to the MRSS distribution shown in Figure 9.7(c) for the 45 cone angle crystal at 7.5 mm from the interface. The eight-fold distribution of the mode at the edge of the section is associated with the eight fold symmetry of the MRSS. 148 a b 149 Figure 9.11 Temperature distribution as a function of external temperature gradient for a 45° cone angle crystal. Thermal gradients in the boron oxide are : (a) 50°C/cm, (b) 100°C/cm, (c) 200°C/cm and (d) 400°C/cm. Temperatures are given in 10 °C. For low gradients isotherms are nearly flat and slightly convex. For larger gradients isotherms are curved and concave. 150 Table 9.2 Effect of Ambient Temperature on Axial Temperature Gradient and Stress Temperature Gradients, °C/cm Boron oxide | Argon | Axial | AMRSS, MPa 50 25 25 0.59 100 50 48 2.55 200 100 84 5.12 400 0 98 11.91 For a temperature gradient of 400°C/cm (Figure 9.11(d)), the isotherms are clearly curved with the curvature increasing near the shoulder. The figure also shows that the axial temperature gradient is higher than the radial gradient. The axial gradients in the crystal for the different imposed gradients are listed in Table 9.2. It can be seen that these gradients are not linearly related. In the same table the largest MRSS (AMRSS), which occurs below the shoulder, is also listed. The relation between the axial gradient and the AMRSS is not linear. The stress distribution in the crystal growing under a thermal gradient of 50°C/cm is shown in Figure 9.12(a) and (b), for a 200°C/cm gradient in Figure 9.13(a) and (b) and for a 400°C/cm gradient in Figure 9.14(a) and (b). Part (a) gives the MRSS, part (b) the MRSS-CRSS (Yield). At 50°C/cm (Figure 9.12(a)) the MRSS has a different stress distribution than at higher gradients specifically the high stress below the shoulder 151 Figure 9.12 Stress contours (MPa) in a C010^ plane for a 45° cone angle crystal grown with a 50 C/cm gradient in the boron oxide, (a) MRSS contours does not show the bump shaped stress distribution below the shoulder in (a). The MRSS-CRSS (Yield) is positive only in a few regions in (b). 152 Figure 9.13 (a) MRSS contours and (b) MRSS-CRSS (Yield) contours for a 200 C/cm gradient in the boron oxide. Units are in MPa. MRSS stress are large below the shoulder in (a). The MRSS-CRSS is greater than zero in most of the crystal in (b). 153 Q b 154 O-O-OO-O-O 6 o ft ft ft 6 9 o ft ft ft cb III o o ft ft a 6 6o ft ft D 6 6 O DO oo IV o o o o o o o o o 6 o o 6 o o 6 o o o o o ooo V o o o o o o o o o\>s o o o o o o os o o o o o o oo \ \ oooooooooooo ooo oooooooooo^ 6 o o OOOOOOOOOOD OOOOOOftftftftftftftftD O O O O ft ft ft ft ft A ft ft A ft ftND OOftftftftftftftOOOOOO D^J \ 6 o o o o ft ft ft ft ft ft ft ft ft ft ft • o o • • • ft ft ft ft o o o o o • • 6 6 o • • • • ft ft o o o o o o o • o 6 o o • • • • ft ft o o o o o o o <!> 6 o • • • • • ft ft o o o o o o o o ? ° • • • • • • ft ft o o o o o o 6 o o • • • • • • • • ft ft o o o o ft o o 1 o • • • • • • D • Q ft ft ft O D O-ft. -ft--ft- .ft--ft- -ft. -ft--ft-• ft--ft. -o- -o- -o- -o- -o-6 Figure 9.14 (a) MRSS contours. (b) MRSS-CRSS (Yield) contours. (c) Slip mode distribution. Gradient in the boron oxide is 400 C/cm. In (a) contours are similar to those obtained for a 200°C/cm gradient. Stress levels have doubled. In (b) only a small area in the seed developed stresses less than the CRSS. In (c) the mode distribution is similar to that shown in Figure 9.5 for a 100°c/cm gradient. 155 disappears. At the higher gradients, Figures 9.13(a) and 9.14(a) the MRSS configuration is similar. The MRSS level increases as the thermal gradient increases. This is shown in part (b) of Figures 9.12 to 9.14. For a gradient 50°c/cm almost the whole crystal section has MRSS-CRSS (Yield) levels below zero. The slip mode which is operative at 400°C/cm is the same to the mode which operates at 100°C/cm as seen by comparing Figures 9.14(c) and 9.5. From this it can be concluded that similar slip modes come from similar stress configurations and this is independent of the value of the constant temperature gradient. Similar slip modes are expected for stress distributions with similar relation among stress components. In addition the symmetry of the MRSS does not change when the thermal gradient is increased from 100°C/cm to 400°C/cm as shown by Figures 9.7(c) and 9.15. 9.1.3 Effect of the Heat Transfer Coefficient Conditions : BG, 100°C/cm ; AG, 50°C/cm In order to determine how sensitive the calculations are to the magnitude of the heat transfer coefficients, the calculations were repeated with values of the heat transfer coefficient increased and decreased by a factor of 1.3 respectively. The results are shown in Figure 9.16 for the temperature profiles at the surface of the crystal. Four profiles are presented ; three correspond to the three different heat transfer coefficients and the fourth corresponds to the ambient temperature. The 156 CO 10] Figure 9.15 MRSS contours in MPa in a cross-section at 7.5 mm from the interface for the crystal shown in Figure 9.14(a). The symmetry is similar to that shown in Figure 9.7(c) for a 100°C/cm gradient. 157 temperature profiles at the surface and centreline, which are not shown, are very similar to those in Figure 9.16. The largest effect of the change in heat transfer coefficient is observed at the seed end. The change in temperature follows the change in the heat transfer coefficient. At the crystal/cone junction the temperature change is small, approximately 5°C. At the seed the effect is more pronounced because the seed radius is small and heat transfer is controlled at the seed surface. In the bulk crystal there is mixed control at the surface and in the bulk material. As a result, a change in heat transfer coefficient does not substantially modify the temperature field. The stress fields are shown in Figure 9.17 and 9.18 for the two heat transfer coefficients h / 1.3 and h X 1.3 respectively, (a) corresponds to the MRSS distribution and (b) corresponds to the MRSS-CRSS (Yield). Comparing the maximum values of MRSS occurring below the shoulder, it can be seen that the stress varies only about 20 * in the whole span of heat transfer coefficient values. The span represents 53 % of the original values at the corresponding temperature. Inside the crystal, changing the heat transfer coefficient has a strong effect on the stress field. Note that the minimum value of MRSS increases from 0.05 MPa to 0.16 MPa. Increasing the heat transfer coefficient does not change the dislocation distribution (Figure 9.17(b) and Figure 9.18(b)). This can be 158 0 I 2 3 Position Relative To Interface (cm) Figure 9.16 Temperature profiles at the surface of 45 cone angle crystals for three (liferent conditions. Curve H corresponds to the temperatures calculated using the original heat transfer coefficient values. In curves H / 1.3 and H x 1.3 the original heat transfer coefficient values were divided and multiplied by 1.3 respectively. 159 Figure 9.17 Stress contours in MPa derived from the temperature field obtained using heat transfer coefficient values 1.3 larger than original values. (a) MRSS contours, (b) MRSS-CRSS (Yield) . 160 Figure 9.18 Stress contours in MPa field obtained using heat 1.3 times smaller than contours. (b) MRSS-CRSS 20 % are observed in the (a) and Figure 9.17(a). derived from the temperature transfer coefficient values original values. (a) MRSS (Yield). Variations of only maximum MRSS values between 161 accounted for by the increase in the thermal gradients in the crystal resulting from the increase in heat transfer coefficient, which leads to two opposite effects. These are the higher thermal stresses and the higher CRSS in the crystal resulting from the lower temperatures. When both effects are considered the net result of the increase in heat transfer coefficient on the dislocation distribution is negligible 9.1.4 Effect of Non-linearity in the Temperature Profile The temperature field in the crystal depends almost entirely on two parameters, the heat transfer coefficient at the crystal surface and the temperature profile in the media surrounding the crystal. These two variables are interrelated and may depend on many factors like heat input into the mold, crystal diameter and gas pressure. However when the temperature profile, the height of the boron oxide layer, the gas composition and the pressure are known, the heat transfer is determined at each point on the crystal surface. This heat transfer coefficient, together with the temperature profile, crystals dimensions, and growth velocity, determine the temperature field inside the crystal. In the previous section it was shown that a 30 * variation in the heat transfer coefficient does not substantially affect the stress field. However it was shown that changing the temperature profile in the surrounding media has an important effect. Increasing or decreasing the thermal gradient markedly changes the thermal stresses. In those calculations the temperature profile in both the boron oxide and the argon gas was assumed to 162 vary linearly with distance from the solid/liquid interface. This may not be the case, as shown by the temperature profiles measured along the crystal surface. In order to determine the importance of non-linearity of the imposed temperature profile, temperatures and stresses are calculated for the four growth conditions listed in Table 9.3 Table 9.3 Effect of Non-linearity of the Ambient Temperature Profile on Stress Temperature Gradients, °C/cm 200, z < 1.0 100, z > 1.0 AT in BO. Run 1 1 2 3 I B203 | Argon | °C 140 100 350 100 350 100 , z < 1.0 50, z > 1.0 50 200 45 , z < 1.0 90 , z > 1.0 180 Note Boron oxide thickness = 25 mm ; argon pressure = 30 atm. ; radius = 20 mm ; length = 10 mm ; cone angle = 30° . z is the distance from the solid/liquid interface. 163 The calculations are divided in two sets 1, 2 and 3, 4 taking into account the difference in temperature across the boron oxide layer. Each pair of conditions gives similar average temperature gradients across the encapsulant. The difference is in the temperature profile. In 1 the gradient is constant, in 2 and 3 the gradient is larger close to the melt, and in 4 the gradient is larger close to the encapsulant/gas interface. The MRSS for the four cases are given in Figure 9.19(a-d). Comparing (a) and (b) it is seen that the relative stress distribution in both cases are similar. The AMRSS below the shoulder for case 2 in (b) is twice as large as the equivalent stress for 1 in (a). The minimum MRSS inside the crystal in (b) is about 20 times larger than in (a). Comparing 3 and 4 in Figure 9.19 the MRSS distributions differ. In (d) the stress concentration occurs along the cone surface rather than below the shoulder as in cases 1 to 3. The AMRSS on the cone surface in (d) is about 1.6 times larger than in (c) for 3. The lowest stress in (c) occurs at two positions, in the cone and below the cone. In Figure 9.19(d) the lowest stress occurs only below the cone. The value of the minimum MRSS in (c) and (d) is the same. The symmetry of the stresses in (001) planes perpendicular to the growth direction is similar for 1, 2 and 3 of Table 9.3 and similar to the symmetry obtained for crystals grown with a 1 cm deep encapsulant and a gradient of 100°C/cm. These symmetry patterns are shown in Figure 9.7(c) and 9.9(c). The symmetry of the stresses for 4, however, is four-fold at sections 5.0 and 164 165 Figure 9.19 MRSS (MPa) contours for a crystal growing under four different conditions given in Table 9.3. (a) Run #1 ; (b) Run #2 ; (c) Run #3 ; (d) Run #4. From (a) to (b) the AMRSS doubles. In (d) the bump shape below the shoulder does not appear. 166 7.5 mm from the interface. At 5.0 mm evidence of two-fold symmetry is observed as shown in Figure 9.20. The MRSS-CRSS (Yield) for conditions 1 to 4 are shown in Figure 9.21(a-d). For 1 the stress over large areas of the crystal is less than the CRSS, for 2 only small areas have stresses below the CRSS. Similar results are obtained for 3 and 4. In 4, however, the region with zero stress at the cone has moved to a position below the cone. When comparing the stresses obtained for a boron oxide layer 2.5 cm thick (Table 9.3) with stresses generated with a 1 cm layer, the MRSS-CRSS (Yield) for 1 is comparable to the distribution obtained using a gradient of 100°C/cm across the encapsulant (Figure 9.4(a)). It is noted that in 1 larger gradients are present, 40 * larger in the crystal and 300 * larger in the cone when compared to the gradients in a 1 cm thick layer. This result shows the importance of suppressing the thermal discontinuity occuring with the 1 cm thick encapsulant. The stress distribution of Figure 9.21(b) for case 2 is comparable to the distribution in Figure 9.13(b) for a 1 cm thick layer. Comparable distributions are also obtained in Figure 9.21(c) for case 3 and Figure 9.4(a) for a 1 cm layer. This shows that for this crystal length, increasing the depth of the boron oxide from 1 to 2.5 cm with the same temperature profile in the ambient does not decrease the dislocation density. In addition it can be observed that the areas with stresses less than the CRSS (Yield) in Figure 9.21(c) are similar to those in Figure 9.21(a). 167 CO 10] Figure 9.20 MRSS (MPa) contours in a (001) plane at a distance of 5.0 mm from the interface in the crystal shown in Figure 9.19(d) (run #4). The four-fold symmetry with a slight two-fold symmetry is observed. 168 169 c d 21 MRSS-CRSS (Yield) contours (MPa) for the four growth conditions shown in Table 9.3 and MRSS contours shown in Figure 9.19 (a) Run #1 ; (b) Run #2 ; (c) Run #3 ; (d) Run #4 . 170 In 1, however, the gradient Is almost twice as large as the average gradient in 3. This shows the detrimental effect on stress of gradients which are not constant. From the results presented in this section it has been shown that the effect of the shape of the imposed temperature profile is very important and that the approximation of a constant gradient in the temperature profile in the boron oxide, as done 19 5 by Dusseaux , gives stresses which are different than the stresses obtained using variable gradients. In addition it has been shown that increasing the boron oxide thickness alone, without considering the changes in the thermal profiles in the surrounding media, does not contribute to a substantial decrease in the stress levels. 9 . 2 Crystal Length Conditions R, 27.5 mm ; B, 21.0 mm CL : a) 13.75 mm ; b) 27.5 mm ; c) 55.0 mm ; d) 82.5 mm ; e) 110 mm . In the previous section it was shown that the ambient temperature profile is a very important variable in determining the stress level in a growing crystal. This is for both the axial temperature gradient and the curvature of the temperature profile. As a result it is very important to use, as input in the model, temperature profiles which are as close as possible to real conditions inside the crystal growth chamber. Measurements 171 of temperature profiles during growth have been reported using the thermocouple arrangement shown in Figure 8.18. The results are shown in Figure 8.19 in which curve 4 gives the ambient temperature adjacent to the crystal surface. The MRSS distribution for the five crystal lengths considered are shown in Figure 9.22(a-e). Comparing distributions shows the following. 1) In all cases high stress levels occur at the centre and the outside part of the crystal. For the shortest crystal, (a) high stresses are also present close to the solid-liquid interface. 2) The location of the maximum (AMRSS) is always at the crystal surface and its position in the axial direction changes with crystal length. For crystal lengths less than the thickness of the encapsulant the position of the AMRSS is below the shoulder. For longer crystals the AMRSS is above the encapsulant-gas surface. 3) The minimum stresses occur at both ends of the crystal and also in a narrow axial band along the crystal, usually half way between the centre and outside part of the crystal. The position of the minimum moves to the crystal surface at both ends. The stress distribution in the radial direction always exhibits a W shape . Close to the ends the valleys in the W are closer to the outside part of the crystal. 173 4) The value of the AMRSS which is representative of the stress level in the whole crystal, increases with crystal length, reaching a maximum value for (c), and slightly decreases for longer crystals. The maximum AMRSS for (c), for a crystal length of 55.0 mm (aspect ratio = diameter/length = 1) is the result of two main factors. First, higher stresses are expected above the encapsulant surface because of the discontinuity in the media surrounding the crystal. This produces a sudden change in heat transfer coefficient and gives a maximum curvature in the temperature profile in the surrounding media. In Figure 8.19 curve 4, this is observed at 30 mm from the interface. As a result of these changes in the temperature field the stresses are high. The second factor contributing to the largest AMRSS for (c) is related to the geometry of the deformation. For the 55 mm crystal length the effect of the thermal transition occurs at the midlength of the crystal. The material in this region is more restrained to deform than the regions close to the ends. This quasi-plane strain deformation gives higher stresses. The contours of the MRSS-CRSS (Yield) for the five crystal of increasing lengths are shown in Figure 9.23(a-e). The results are similar to the MRSS considered previously. In addition it can be observed that at the shortest length for (a) there is a small area in the cone as well as below the cone where the MRSS is less than the CRSS (Yield). For the two largest crystals ((d) and (e)) 175 the top of the crystal does not develop stresses higher than the CRSS (Yield). When the CRSS (MB) stresses for dislocation generation in undoped GaAs is considered, the contours and stress values are very similar to those obtained for the MRSS (Figure 9.22). The contours of the MRSS-CRSS (MBTe) in Te-doped material are similar to those obtained when the yield stress is considered and shown in Figure 9.23. The operative slip mode of the MRSS as a function of position in the crystal is shown in Figure 9.24(a-e). The distribution indicates that the slip mode operating may not be associated with the stress level. The slip mode alternatively may be related to specific regions in the crystal. The most frequent modes are III and IV. Mode III appears in the area of maximum stress above the encapsulant surface and at the centre of the crystal. It also appears in the cone and in the central region at a distance from the interface greater than 27.5 mm. Mode IV preferentially appears in the following regions (a) close to the interface (b) above the region of maximum stresses and (c) below the shoulder. The other slip modes are less frequent. 9.2.1 Effect of crystal Length on Stress Symmetry The effect of crystal length on stress symmetry is presented in two ways. First, different cross-sections are followed during growth and the symmetry changes are described. Following this the symmetry at each crystal length is considered and the changes with length are reported. 176 •-D-O-D-O-O-B-B -B-B-B-B-B -fl-B-B-B-B -O-O-O-O-O-O-O-O-B-B-^-^-^-O-O /••••••••••••••••••••••••aaaoaoooooooo^^o^i /.ODaQOaaoaDiiiiiiniaQQGQaQDQDoooooooo^aa^ o0o44 444 4^D««t**««***«««o44444<l**«*ooooooo4 _ /o 0ooo44oooooooooooooooooooooBBBBBBaaaaoaao« /°°0OO0OOO0000O0O0OO0O00o0O000OBBBDDODaDDDD0» o0°0ooooooooooooooooooooooooooooiDDDDoaoaaDoi Oo0oooooooooooooooooooooooooooooarjarjaaaoaaa« /oooooooooooooooooooooooooooooooooooaaoDDDOOooi •-• oooooooooooooooooooooooooooooooooooooooooooooi o-o-o-o-o-o-o-o-o-o-o-o-o-o-o- o-o-o-o-o-o-o-o-o-o-o-o- o-o-o- o-o-o-o- oo-o-oo- o-o-o-o- o- o-c-o-6 D-Q-O-O-O /*••••• o o • • a a 'o _ o o o o o w O n O /Ooo00000 p0g0°°ooo oS°ooOoo OO /oo°o°° 0 0 00 •"•00000000 OO 0-0-000-0-0-0-0-0 -0-0 B-B-B-B-B-B-B-B-B-O-O-O-O-O-O-O-O-O-O-••••••••••aoaoooooo • •ooddoooaoorjoooooo OOOOOOOO OOIIIIIIIQQ OOOOOOOOOOIIIIDEIDD OOOOOOOOOOOIIOODOD OOOOOOOOOOOODDDODA OOOOOOOOOOOOOQDDQO OOOOOOOOOOOOOOOOOO 0-0-0-0-0-0-0-0-0-0-0-00-0-0-0-0-0-a-o-4-4-a-o 00 o44o4 O O O <J • • • oo^aac^ • a • a • a • o a • • a • f OOOOOOf •0-0-0-0-0-0-0 a-a-a-D-o-o-o-o-D-o-o-o-o-o-o-o ^ -4 <a-o Q/D DQGQQOQD000000000404 /Q4 ••••••••oooooooooaa* p0m<4<4M44M •••••••• oaa • ^"QOOOOOIIIIIIIIQDDDDDDI 0Oa00O00000IIIIIQDCiaciclDC1* /oQ°0ooooo oooiiiaoaaaaQaoi /OOOOOOOOOOOOOOIIIODDODQQQI •-o oooooooooooooooooomioooi 6-0 -0-0-0-00-0-0-0-0- 0-0-0-0-0-0-0- o-o-o-o- 0-0-0-0-0-0 •-a-0-0-0 -O-0-4-4 -a-o Ja ••D0000004 o0oooniiiDaaQ« 0^OoOOBBBBBQDDn» /Oo°OOOOBBBBBBBD» •-• 0000000000000001 6-000-0000-00-0-0-0-0-0-0-0-6 D-O-O^-O-O - - •— •— 1— 1— — •— o o o o 00 I — I— -I— I— I-I— -I— I-'• O O • B /••D» • B • Ooo«» B D /0QQ««BB /JOBOOBOBB. /JQOBOOBO OB • •-• OoOBOOO O OO O O OOOO-OOO-O-O-O-O-O • > O o o o — -I— — — I— 00 — I-I— I-CO 1—4 X CO •*-> £ 4-> to E CO c >> cu in u 1—4 0 c-CM CO • 4-> ^-^ CO E >> •—• E Si 0 O cu O > --1 -1-4 i-i E E ^—-cu in J= t> 4-> CO --(1 1-1 O £ tM .—. E CO ' IO CO to 00 as CO co s CM • . E cu -— E cu +-> u lO 3 to t> 0 E CM b E cu CO C O •1-1 O -1-1 • e CO c m 3 a 2 •rt 1—( O T3 —H X CO 03 CO U CO C7> CU Si 3 to •—) 177 Considering the stress symmetry during growth four sections fixed in the crystal are examined at 8.25 mm, 49.5 mm, 77.0 mm and 104.5 mm from the cone. The symmetry and stress levels in these cross-sections are considered in relation to dislocation formation. In the model the dislocation density formed at any given time and position is assumed to be proportional to the MRSS-CRSS (Yield). However, the stress level changes during growth. It is therefore important to introduce two new considerations. The first is related to hardening. The effect of hardening is to decrease the effective stress in the crystal due to dislocation interactions. The decrease in stress is proportional to the square root of the dislocation density, i.e. A x /~Ii, where A is the interaction constant and n is the 19 2 dislocation density. According to Vakhrameev et al . A varies - 2 from 2 to 4 x 10 MPa x m. The magnitude of stress involved in - 3 - 2 the interaction varies between 4 x 10 MPa and 4 x 10 MPa for 2 4 2 dislocation densities of 10 to 10 /cm These stresses are negligible with respect to the thermoelastic stresses developed during growth. Therefore it may be concluded that the hardening effect is minimal for normal dislocation densities. The second consideration is partially a consequence of the first and is related to the increase in dislocations after each step during growth. Because there is no hardening effect it is assumed that the dislocation density in a given section increases proportionally to the difference in MRSS-CRSS (Yield) values between two steps. The final density is given by the maximum stress reached at each point. 178 The specific observations in the four sections follow. In all cases the MRSS-CRSS (Yield) is considered. 1) Section at 8.25 mm from the cone. CL : (a) 13.75 mm ; (b) 27.5 mm ; (c) 55.0 mm; (d) 82.5 ram. The stress distibutions at the four steps are shown in Figure 9.25(a-d). The distribution shown in (a) corresponds to an early stage of growth when the crystal is contained by the encapsulant. The stress level in most of the crystal section is less than "1 MPa. The symmetry changes depending on the position in the section. Circular symmetry is obtained in the centre which changes to four-fold, eight-fold, circular and finally eight-fold on moving from the centre to the edge of the crystal. In Figure 9.25(a) the dislocation density is low and has the same symmetry. In (b) the stress level at the edge is about 6 times higher than in (a). The changes are not as important at the centre where the stress has not quite doubled. In a ring midway between the centre and the edge the stress has substantially increased. The symmetry is almost circular except at the edge where the eight-fold symmetry is stronger than in (a). In (c) the stress at the periphery has decreased 30 * with respect to (b). At the centre the stress has doubled. In the central area the symmetry has taken a more definite four-fold pattern which appears again in the ring midway between the centre and the edge. In this case there is a slight two-fold symmetry which can be seen as an elongation of the minimum stress area in the [110] direction with 179 a 180 181 182 gure 9.25 MRSS-CRSS (Yield) (MPa) in a (001) plane at a distance of 8.25 mm from the cone at four crystal lengths. (a) 13.75 mm ; (b) 27.5 mm ; (c) 55.0 mm ; (d) 82.5 mm. Crystal radius 27.5 mm. 183 respect to the [110] direction. In (d) the stress level has decreased in the whole wafer and therefore no more dislocations are generated with further growth. At the end of growth the final dislocation density distribution in this wafer will be determined by the maximum stress reached during growth. At the edge this stress is reached in (b). At the centre and midway region the largest stress is reached in (c). From these maximum values it may be predicted that the wafer will show a W-shaped dislocation distribution along a diameter. The branches of the W will be more than twice as high as the centre. The eight-fold symmetry at the edge will be hardly noticed since it occurs in a narrow ring with dislocation variations of the order of 10 % . The four-fold symmetry will be also very weak. 2) Section at 49.5 mm from the the cone. CL : (a) 55.0 mm; (b) 82.5 mm . The stress distributions for cross-sections at 49.5 mm from the cone is shown in Figure 9.26(a) and (b). Figure 9.26(a) corresponds to a stress distribution close to the interface. The following characteristics can be distinguished. The central part has four-fold symmetry and the rest of the surface has almost circular symmetry. The stress level is the same at the centre and the edge, and the minimum occurs close to the edge. In (b) the stress level in the central region has increased by 50 % and at the edge the stress has doubled with respect to (a). The symmetry in (b) is almost circular throughout the 184 185 Figure 9.26 MRSS-CRSS (Yield) (MPa) in a (001) plane at a distance of 49.5 mm from the cone at two crystal lengths. (a) 55.0 mm ; (b) 82.5 mm. Crystal radius 27.5 mm. 186 section. The final maximum dislocation density in this wafer may be determined by the stresses developed in (b). In this case a W-shaped distribution is predicted with the center of the W of the same height as the branches. Since a considerable number of dislocations are generated in (a) with four-fold symmetry, this will be present in the final distribution. When the dislocation density distribution predicted in this section is compared with the final dislocation distribution given in (1) for a section close to the cone, the predicted dislocations far from the cone will be 10 % lower in the outer region and 50 % higher in the centre. 3) Section at 77.0 mm from the cone. CL (a) 82.5 mm; (b) 110.0 mm . The stress distributions for a wafer 77 mm from the cone are shown in Figure 9.27(a) and (b). Similar patterns as in 2) are observed in this case. The differences are that in (a) the four fold symmetry is expanded and in (b) there are four minimum stress regions midway between the centre and the edge in the <100> directions. 4) Section at 104.5 mm from the cone. CL : 110.0 mm. cone and The only cross-corresponds to it is close to section available at this distance from the the longest crystal shown in Figure 9.21(e) the solid/liquid interface. The stress 187 a 188 Figure 9.27 MRSS-CRSS (Yield) (MPa) in a (001) plane at a distance of 77.0 mm from the cone at two crystal lengths, (a) 82.5 mm ; (b) 110.0 mm. Crystal radius 27.5 mm. 189 distribution in this section is very similar to the distribution shown in Figure 9.27(a). If growth is completed when this length is reached the dislocation distribution at the tail end will have the features observed in Figure 9.27(a). These are 1) an expanded four-fold symmetry, 2) a dislocation distribution in the radial direction with a W shape across the diameter having a minimum very close to the edge, a wide peak in the centre, and branches equally high. The average dislocation density in this wafer will be half of the dislocation density for a wafer considered in 3) at 27.5 mm. If instead of following the stress distributions during growth in specific sections, the distributions of stresses are observed along the crystal at each of the five lengths considered, the following patterns can be observed. 1) The eight-fold symmetry at the edge of the sections is always present. This symmetry is associated with an eight-fold distribution of the slip mode of the MRSS. 2) In the radial direction, the stress distribution always shows a W shape. 3) The four-fold symmetry is stronger close to the interface between 5.5 mm and 11.0 mm. It also appears at the top of the crystal near the cone. 190 4) A slight tendency to a two-fold symmetry when present is always accompanied by a four-fold symmetry. 5) Far from the ends, at least 13.75 mm, the stress symmetry is almost axisymmetric with slight four-fold symmetry at the center and eight-fold symmetry at the edge . 6) For crystal lengths larger than 55.0 mm, cross-sections far from the interface have the minimum stress levels in the <100> directions rather than in the <110> directions. The area where this minimum occurs is very smal1 . 9.3 Crystal Radius Conditions : R, 40.0 mm ; CA, 30° ; B, 21.0 mm ; AP, 30 atm ; CL (a) 10.0 mm ; (b) 40.0 mm ; (c) 80.0 mm ; (d) 100.0 mm . The effect of increasing the crystal radius from 27.5 mm to 40 mm on internal crystal stresses and dislocation distributions is determined assuming that the temperature profile in the medium surrounding the crystal has not changed when the radius is increased. In this case the temperature profile is given in Figure 8.19 (curve 4). Values for the MRSS at the four crystal lengths considered are shown in Figure 9.28(a-d). The MRSS distributions follow the 191 same general patterns as those observed with the smaller crystal radius, with two main differences. The first is that the maximum stress, considering the whole growth process, occurs when the crystal is 40.0 mm long in (b), with an aspect ratio of 1 rather than 2 as in the smaller crystal. This difference is consistent with the explanation given in the last section. The maximum stress usually occurs above the encapsulant surface. If this position coincides with the midlength position in the crystal there is an extra contribution to the stress due to the quasi-plane strain conditions prevailing there. The second difference is the higher stress level for the larger crystal. In this case increasing the radius 45 % increases the stresses by 40 % indicating that the MRSS increases proportionally with the radius. The effect of increasing the radius on stress levels has been explained in the literature by considering the effect of increasing the radius on the thermal field. As pointed out previously, the results for the effect of radius on temperature fields are contradictory. This could result from the limitations of the solutions employed to study this effect. In the present investigation the effect of radius on thermal gradients is first considered as a possible cause for the observed effect on stresses. The comparison can be made using two criteria. It can be made for two crystals of same aspect ratio, or between two crystals with the same length. In this investigation the second method is chosen. (The conclusions obtained with both methods are similar). For the comparison crystal lengths for which the 192 maximum stresses occur are chosen. For the 27.5 mm radius crystal the length is 55 mm. For the 40.0 mm radius crystal the length is 40 mm. Because the lengths are different, another length for the 40 mm radius crystal is also considered, specifically 80 mm length. For the three crystals, gradients in two directions are considered ; axial gradients along the crystal axis and average radial gradient. The calculated gradients are shown in Figure 9.29(a) for the axial gradients and (b) for the radial gradients as a function of distance from the interface. Curve (1) corresponds to the 27.5 mm crystal and curves (2) and (3) for the 40 mm crystal. It can be observed in the figures that the curve (3) is below curve (1) in (a) and curve (1) is between (2) and (3) in (b). From these observations it can be concluded that the thermal field is not influenced by changes in radius and that the large change in stress observed cannot be accounted for by the change in thermal field. An alternative explanation for the effect of radius on stress is related to the specific geometry of the crystal. In the case of cylindrical bodies the radial gradient has a different effect on stresses than in cubic geometries. In cylinders a constant radial gradient can develop stresses ; in cubes it cannot. A radial gradient in a cylinder generates incompatibilities in the strain components for free expansion. From the present results it is concluded that the same radial gradient in two cylinders of different radius creates larger incompatibilities in a crystal with larger radius. This is more clearly seen when it is considered that the effect of thermal 194 Figure 9.29 (a) Axial along the crystal axis and (b) radial thermal gradients as a function of distance from the interface for two crystal radius. Curve (1) for a radius of 27.5 mm ; and curves (2) and (3) for a radius of 40.0 mm. Crystal lengths are (1) 55.0 mm ; (2) 40.0 mm and (3) 80.0 mm. 195 strain is included as an initial strain calculated as a free expansion from a reference temperature. This free expansion (contraction in this case) increases with increasing distance from the centre. The degree of incompatibility is expected to follow the expansion, which in turn will generate larger stresses . The effect of a constant radial gradient G on the stress in cylinders of different radii can be quantitatively evaluated using the plane strain approximation. The stress components for a temperature field T = -Gr G a E Or - (r - r ) 3 1 - V <2r - r0) a E 1 - v Oz = G (r - 2/3rQ) a E l - v It can be seen that the stress components increases linearly with the radius r . The MRSS-CRSS (Yield) contours for the four lengths considered are shown in Figure 9.30(a-d). The results shown are similar to those obtained with the smaller crystal, with areas having stress levels below the CRSS in (a) and (d). The distribution of the MRSS-CRSS (Yield) for the four crystal 197 lengths show contours and levels which are very similar to the MRSS contours. The distribution of MRSS-CRSS (MBTe) are similar to those in Figure 9.30(a-d). 9.3.1 Effect of Radius on Stress Symmetry The stress symmetry in crystals with four different lengths has been examined. The cross-sections studied correspond to the same relative positions considered in the crystals with smaller radius. As before, the analysis was done in two ways namely following a fixed cross-section in the crystal during growth and analysing the symmetry distribution in the crystal for each crystal length. From the second analysis, the conclusions are similar to those obtained for the smaller radius. From the first analysis by following a cross-section and comparing the results with those obtained for the smaller radius crystal, the following conclusions were reached. There is a correlation of symmetry with distance from the cone between the two crystal radii. This correlation is given in Table 9.4 which was obtained in the following way. Consider the sections at 77 mm (2.8 x R) for the 27.5 mm crystal and 72 mm (1.8 x R) for the 40 mm crystal. Both sections have very similar stress symmetries during growth. In this case the stress symme tries shown in Figure 9.27(a) and (b) corresponding to a relative distance from the cone of 2.8 times the radius is reproduced in the larger crystal but at a relative distance of 1.8 times the radius. Similar correlations are obtained following other section at the same distances from the cones for the two radius. 198 Table 9. 4 Table of Symmetry Correlations Radius [mm ] Distance from Relative Distance | Shoulder [mm] | from shoulder | Figure 27 40 . 5 . 0 8.25 0.3 12.0 9 . 25 27 40 . 5 . 0 77.0 2.8 72.0 1. 9 . 26 27 40 . 5 . 0 104.5 3.8 112.0 2. 9 . 27 Because this correlation is with actual distance, the effect of the thermal field is expected to predominate over the effect of the boundary conditions. If the symmetry is determined by the thermal field alone, there should also be a strong correlation between the symmetries and the actual distance from the interface. This correlation is not observed. On the contrary, for long crystals there is correlation of symmetries but with the relative distance from the interface for same crystal lengths. The correlation with actual distance from the cone, however, suggests that the thermal history of the wafer is important in determining the stress symmetry. This would results in sections passing through the same thermal fields having the same symmetry history regardless of the crystal radius. 199 9.4 Growth Velocity Conditions R , 27.5 mm CL, 55.0 mm B, 21 mm Temperature profile as in Figure 8.19, curve 4, Velocity (a) 0.0001 cm/s, (b) 0.001 cm/s , (c) 0.01 cm/s To determine the effect of changes in the growth velocity on the thermal and stress fields, the calculations were repeated with growth velocities of 0.0001 and 0.01 cm/s. The temperature fields for the three velocities considered are shown in Figure 9.31(a-c). Comparing Figure 9.31(b) and (a), it is noted that decreasing the velocity has little effect on the temperature distribution. Increasing the velocity (Figure 9.31(b) and (c) ) significantly increases the axial and radial temperature gradients in the crystal. The results indicate that at velocities of 0.001 cm/s and lower the thermal system is at steady state, and effectively independent of velocity. Above 0.001 cm/s this is not the case, primarily due to the value of thermal diffusivity used (0.04 cm/s) which is comparable to the velocity. The specific effect of growth velocity on the thermal fields is complex. In the quasi steady state approximation, the temperature field changes only as a result of the movement of the solidifying interface ; the initial transient is neglected. In the present calculations, and independent of the heat transfer at the crystal 200 201 C Figure 9.31 Temperature fields for three growth velocities (a) 0.0001 cm/s, (b) 0.001 cm/s and (c) 0.01 cm/s. Temperature given in 10 °C. Little change is observed from (a) to (b). From (b) to (c) gradients have increased. Radius 27.5 mm, length 55 mm, encapsulant thickness 21 mm. •-%f\L f'U 202 surface, there are two limiting cases. At low velocities steady state fields are expected ; at high velocities uniform temperature fields are expected with the temperature the same as the moving boundary. For comparable values of velocity and diffusivity any qualitative analysis must include the fact that the external surface is cooling following Newton's law of cooling. In this case the situation is complicated by the fact that there is mixed control in the bulk and the surface of the crystal with a Biot number which has values close to one. The MRSS contours corresponding to the three thermal fields shown in Figure 9.31(a-c) are given in Figure 9.32(a-c). Comparing the MRSS values at 0.0001 cm/s with those at 0.001 cm/s, the differences are small ; approximately 2 % for the highest values of the MRSS and 4 % for the lowest. Comparing 0.01 cm/s with 0.001 cm/s, the highest values of MRSS have approximately doubled and the lowest values more than doubled. In addition, the area close to the interface with minimum MRSS disappears as the velocity increases. 9.5 Thermal Conditions The temperature distribution in the growing crystal is dependent on the thermal environment surrounding the crystal and the crystal geometry. The effect of the thermal environment is modelled using the temperature measurements reported by Grant et 218 al. as shown in Figure 8.19, curve 4. 204 C Figure 9.32 MRSS-CRSS (Yield) (MPa) for the three temperature fields shown in Figure 9.31(a-c) corresponding to three growth velocities. (a) 0.0001 cm/s, (b) 0.001 cm/s and (c) 0.01 cm/s. In (a) and (b) the stress fields are similar. In (c) the stress distribution and stress values change. 205 The thermal profile for the different thicknesses of boron oxide is modelled assuming that the temperature at the top and bottom of the oxide is independent of thickness for constant power input. This makes the temperature gradient across the oxide linearly proportional to the thickness, as demonstrated 14 6-148 experimentally It is also assumed that reducing the axial gradients in the chamber by increasing the power input or increasing the boron oxide thickness changes the magnitude of the local temperature but does not change the shape of the temperature profile. This means that only the coefficients in the polynomials fitting the original profile are proportionally reduced by a given factor for the new growth conditions. The boron oxide thicknesses considered are 21, 40 and 50 mm the latter being the maximum thickness currently used experimentally. For each thickness, several temperature gradients are considered. The crystal radii examined are 27.5 mm and 40.0 mm. 9.5.1 Radius 27.5 mm 9.5.1.1 Boron oxide thickness 21.0 mm Average gradients (a) 66°c/cm, (b) 33°C/cm, (c) 17°C/cm, (d) 8°C/cm The temperature profiles considered are given in Figure 9.33. Five profiles are shown. The profile labelled G with the 206 Figure 9.33 Temperature profiles along the crystal surface in the environment surrounding the crystal employed in the calculations. The profiles are derived from Figure 8.19, curve 4, reproduced as curve G. The values of gradient are averaged in the crystal length considered. Encapsulant thickness 21 mm. 207 steepest gradient, is the same as shown in Figure 8.19, curve 4, which was determined experimentally. The results for the MRSS-CRSS (Yield) for four temperature profiles are shown in Figure 9.34(a-d). The stress contours for the temperature profile G were shown in Figure 9.33(c). Comparing Figure 9.34(a-d) it can be noted that the stress distribution patterns are similar in the four cases. The stress levels, however, decreases considerably with the average gradient. As a result of the lower stress levels the area with zero stress gradually increases from (a) to (d). The regions with zero stress start to grow from the regions with lower stresses. These regions are in the seed, shoulder and above the interface edge. At the lowest gradient in (d) most of the crystal has stresses which are less than the CRSS (Yield). The two remaining regions with non zero stresses are located where the highest stresses usually appear ; in the crystal surface above the encapsulant and in the crystal axis. 9.5.1.2 Boron oxide thickness 40.0 mm AVG (a) 58 cm/s, (b) 29 cm/s, (c) 19 cm/s B 50.0 mm AVG (a) 50°c/cm. (b) 25°C/cm, (c) 12.5°C/cm. 209 c d Figure 9.34 MRSS-CRSS (Yield) (MPa) for four temperature profiles with average gradients (a) 33 C/cm, (b) 17 C/cm, (c) 11 C/cm and (d) 8 C/cm. Radius 27.5 mm, length 55 mm. Encapsulant thickness 21 mm. 210 The temperature profiles for both encapsulant thicknesses are shown in Figures 9.35(a) and (b). In (a) for the 40 mm encapsulant and in (b) for the 50 mm. In each case three temperature profiles are considered, the average gradient for each profile is also given. For comparison the original profile for the 21.0 mm encapsulant is also shown. The results for the MRSS-CRSS (Yield) for the two encapsulant thicknesses are shown in Figures 9.36(a-c) for the 40 mm thickness and Figure 9.37(a-c) for the 50 mm thickness. Parts (a) to (c) corresponds to the three different thermal profiles shown in Figure 9.35(a) and (b). To observe the effect of increasing the encapsulant thickness only the three figures to be compared are : 9.23(c) for the 21 mm, 9.36(a) for the 40 mm and 9.37(a) for the 50 mm thickness. It can be noted that increasing the boron oxide thickness changes the relative stress distribution. The changes are the extended low stress region close to the interface and the shift of the stress concentration to a position above the encapsulant surface. For thick encapsu1 ants, the large and uniform low stress region in the low part of the crystal has a stress distribution in a radial direction which maintains the W shape with extended valleys and sharp peaks in the centre and edge of the crystal. In the case of the 50 mm thickness in Figure 9.37(a) the stress concentration region below the shoulder is small because the encapsulant is only 5 mm shorter than the crystal length. 211 Figure 9.35 Temperature profiles used in the calculations for two boron oxide thicknesses. (a) 40 mm, (b) 50 mm. The profiles are derived from curve G. The values of gradients given are averaged along the crystal length. 211 Figure 9.35 Temperature profiles used in the calculations for two boron oxide thicknesses. (a) 40 mm, (b) 50 mm. The profiles are derived from curve G. The values of gradients given are averaged along the crystal length. 212 213 Figure 9.36 MRSS-CRSS (Yield) (MPa) for three average gradients. (a) 58°C/cm, (b) 29°C/cm, (c) 19°C/cm. Radius 27.5 mm, length 55 mm, boron oxide thickness 40 mm. 214 215 Figure 9.37 MRSS-CRSS (Yield) (MPa) for three average gradients. (a) 50°C/cm, (b) 25 C/cm, (c) 17°C/cm. Radius 27.5 mm, length 55 mm, boron oxide thickness 50 mm. 216 The absolute stress also changes with an increase in the encapsulant thickness. Doubling the thickness reduces by half the maximum stress . When thermal profiles with lower gradients are considered for both encapsulant thicknesses, the region in the crystal with zero MRSS-CRSS (Yield) expands from the low stress regions close to the interface as the gradient decreases. The largest zero stress region is obtained in Figure 9.37(c) for an encapsulant thickness of 50 mm and a gradient of 12.5°c/cm. It is noted that similar surface areas were obtained with a 21 mm thickness for a gradient of 8°C/cm. This shows that independent of encapsulant thickness, a critical gradient of about 10°C/cm should be achieved in order to obtain large areas with stresses less than the CRSS (Yield). From the practical point of view, a thicker encapsulant is preferred for the following reasons. To get the desired 10°C/cm level, the thermal gradient should be reduced eight times with the 21 mm thickness and only three times with the 50 mm thickness. This is because of the reduction of gradients when increasing the thickness. In addition in low gradient environments large areas of crystal are exposed to high temperatures. Without proper encapsulation there is decomposition of GaAs, resulting in the formation of liquid Ga drops which damages the crystal and leads to non-stoichiometric melts which is also detrimental. 217 9.5.2 Radius 40.0 mm 9.5.2.1 Boron oxide thickness 21 mm AVG : a) 14°C/cm, (b) 7°C/cm The results for the MRSS-CRSS (Yield) with a boron oxide layer thickness of 21 mm are shown in Figure 9.38(a) and (b). Part (a) corresponds to the profile labelled G/4 (14°C/cm) and (b) to the profile G/8 (7°C/cm) in Figure 9.33. It is noted that the average gradients in these profiles are lower because of the longer crystal considered. Comparing Figure 9.38(a) and (b) with Figure 9.34(c) and (d) for the 27.5 mm crystal, it can be noted that in the case of the 40 mm crystal, the relative zero MRSS-CRSS (Yield) area for the same profile is not as large as in the case of the 27.5 mm crystal. In the 40 mm crystal, however, the top of the crystal in Figure 9.38(b) shows zero stress. This is due to the difference in length between the two crystals. 9.5.2.2 Boron oxide thickness 40 mm AVG : (a) 46°C/cm, (b) 26°C/cm, (c) 15°C/cm, (d) 9°C/cm. The MRSS-CRSS (Yield) contours for the four gradients are shown in Figure 9.39(a-d). Note that the profile with an average gradient of 9°C/cm is not shown in Figure 9.35(a). The 9°C/cm average gradient in the encapsulant and gas is five times smaller than the original temperature profile labelled G in Figure 9.33. Comparing the results of Figures 9.39(a-d) with those in Figure 9.36(a-d) for the 27.5 mm crystal it can be observed 218 Figure 9.38 MRSS-CRSS (Yield) (MPa) for two (a) 14°C/cm, (b) 7 C/cm. Radius boron oxide thickness 21 mm. average gradients. 40 mm, length 80 mm, 220 c d Figure 9.39 MRSS-CRSS (Yield) (MPa) for four average gradients. (a) 46°C/cm, (b) 26°C/cm, (c) 15°C/cm, (d) 9°C/cm. Radius 40 mm, length 80 mm, boron oxide thickness 40 mm . 221 that the concentration of stresses in the larger crystal occurs at the midlength instead of the top of the crystal as in the 27.5 mm crystal. In addition it is observed that to obtain similar relative areas with zero MRSS-CRSS (Yield), the gradients in the environment surrounding the 40 mm crystal should be further reduced. 9.5.2.3 Boron Oxide thickness 50 mm, AVG : (a) 43°C/cm, (b) 14°C/cm, (c) 7°C/cm. In order to determine if the stress level has reached a maximum value considering the whole growth period, stress calculations are made for a 40 mm crystal with a length comparable to the encapsulant thickness. The crystal length is 52 mm and the average gradient is 50°C/cm. The results for MRSS-CRSS (Yield) are shown in Figure 9.40. It is observed that the stress concentration occurs below the shoulder. In Figure 9.41(a) the distribution for the same stress for a crystal length of 80 mm is shown. In this case the stress concentration occurs above the encapsulant surface. The stress value in the longer crystal is higher than the largest stress value in the shorter crystal. From these results and those obtained with the 27.5 mm crystal radius it is possible to generalize the finding to most of the situations in LEC growth. The largest stress at a given crystal length occurs above the encapsulant surface. If this 222 Figure 9.40 MRSS-CRSS (Yield) (MPa) for a crystal length of 52 mm comparable to the boron oxide thickness of 50 mm. Radius 40 mm. Average gradient 50°C/cm. 223 224 Figure 9.41 MRSS-CRSS (Yield) (MPa) for three average gradients, (a) 43°C/cm, (b) 14°C/cm, (c) 7°C/cm. Radius 40 mm, length 80 mm, boron oxide thickness 50 mm. 225 position is half-way along the crystal length, the stress is the largest in the whole growth period. The stress distributions for the three gradients considered are shown in Figure 9.41(a-c). As in the case of the 40 mm encapsulant, the gradient should be further reduced to obtain similar relative areas with zero stress when comparing with the 27.5 mm crystals distributions. The evaluation of the results presented in this section are summarized in Table 9.4. In this table the largest value of MRSS-CRSS (Yield) is given as a function of boron oxide thickness, gradient of the thermal profile and crystal radius. The stress values correspond to crystals with an aspect ratio of 2. For each condition the relative stress increment due to an increase in radius is also reported. It is clear from the tabulated values that the stress decreases as more favourable growth conditions apply, i.e. thicker encapsulant and lower gradient. It is also observed that the relative stress change due to an increase in radius increases as the stress level decreases (or more favourable conditions are present). In the case of the thickest encapsulant and the lowest gradient used in the calculations, the stress increases 143.7 % when increasing the radius from 27.5 mm to 40 mm. This shows that the effect on stresses with increasing the radius is not linear. A comparison of the stress contours for all conditions listed in the table shows that similar stress distributions are Table 9.4 Effect of Thermal Conditions on Stresses Boron Oxide AMRSS - CRSS (Yield) MPa Thickness | G | 1 % [mm] R = 2.75 cm | R = 4.0 cm change G 7.02 9.65 37.46 G/2 3.76 21.0 G/4 1.78 2.76 55.0 G/6 1.03 G/8 0.65 1.02 56.0 40 . 0 G G/2 G/3 G/5 4 . 39 1 . 97 1 . 22 6 . 77 3.21 2.01 1 . 02 54 . 2 62 . 9 64 . 7 50 . 0 G G/2 G/3 G/5 3 . 26 1.14 0 . 64 5 . 61 1 . 56 0 . 47 72.08 143.7 227 obtained for similar values of stress. For instance, similar areas with zero stress are given in the crystals with the following conditions 1) B = 21 mm, G/6, r = 2.75 mm ; 2) B = 21 mm, G/8, r = 40 mm ; 3) B = 40 mm, G/5, r = 40 mm. 9.6 Gas Pressure and Composition Conditions : AP, 2 atm ; R, 27.5 mm ; CL, 55 mm ; B, 21 mm ; Temperature profiles as in Figure 9.33 ; AVG : (a) 66°C/cm, (b) 33°C/cm, (c) 17°C/cm, (d) ll°C/cm. The effect of gas pressure is considered in the model through the effect of pressure on the convective part of the heat transfer coefficient between crystal and gas. In the equations used for the numerical evaluation, the convective heat transfer coefficient changes with the square root of pressure. The results for the MRSS-CRSS (Yield) are shown in Figures 9.42(a-d) for four gradients. The effect of pressure on the four gradients can be analysed by comparing the stress values with the results obtained for a pressure of 30 atmospheres. For this Figure 9.42(b-c) should be compared with Figures 9.34(a-c). It is observed that the stress configuration for the four gradients at 2 atmospheres is similar to the stress configuration for the same 229 c d .42 MRSS-CRSS (Yield) (MPa) for four temperature profiles with average gradients. (a) 33°c/cm, (b) 17°C/cm, (c) 11 C/cm and (d) 8 C/cm. Radius 27.5 mm, length 55 mm. Encapsulant thickness 21 mm. Argon pressure 2 atm . 230 imposed temperature profiles at a pressure of 30 atmospheres. At the largest gradient considered, the stress decreases only 3 % from 30 to 2 atmospheres. For the lower gradients the decrease is in the order of 20 %. For the lowest gradient considered the area with zero MRSS-CRSS (Yield) in Figure 9.42(d) increased about 25 % following the decrease in the largest stress value. The results presented here show that the effect of pressure is not as important as might be expected. The weak effect of pressure observed in the present calculations when a stronger effect is expected, can be accounted for in the following way. In the model the effect of pressure is included as affecting the convective component of the heat transfer coefficient as the square root of pressure. Reducing the pressure from 30 to 2 atmospheres reduces the value of convective heat transfer coefficient by 74 % . At low pressures the convective part is less than 10 % of the total value of the heat transfer coefficient. In this case the maximum variation in the total coefficient associated with the reduction in pressure will be less than 30 * It was shown in Section 9.1 above, that a 30 % change in heat transfer coefficient does not substantially affect the stress level in the crystal. This explains why the effect of pressure in the present calculations is not significant. A stronger effect of pressure is expected because pressure may markedly affect the temperature profile of the media surrounding the crystal. In the present calculations, it was assumed that changing the pressure does not alter the temperature 231 of the environment. This may not be the actual situation. When the pressure changes, not only the heat transfer coefficient at the crystal-gas interface changes but also the efficiency in heat transfer in the gas is altered. This higher efficiency will be manifested by a larger heat transfer coefficient at the boron oxide-gas interface. The net effect will be to decrease the temperature of the boron oxide surface increasing the gradient across the encapsulant and probably in the gas. The change in heat transfer coefficient at the boron oxide-gas surface can be qualitatively described, assuming a free convection mechanism applies at this surface. In this case the heat transfer coefficient associated with this process can be written as I/O h = Nu k/1 = 0.14 (Gr Pr) ' (k/1) (9.1) where Nu, Gr and Pr are the Nusselt, Grashof and Prandl numbers respectively. "k" is the thermal conductivity. "1" is the characteristic length of the horizontal heated surface from which heat is extracted. If it is assumed that the gas behaves like an ideal gas, the pressure dependence is introduced in the Grashof number through the density. In this case the heat transfer coefficient will depend on pressure with an exponent 2/3. For this pressure dependence the heat transfer coefficient is reduced by 83 % when depressurizing the chamber from 30 to 2 atmospheres. At the encapsulant-gas surface this change is more important than similar changes of the convective heat transfer coefficient at 232 the crystal-gas surface since in the first case convection is the main heat transfer mechanism. This is because boron oxide is transparent to radiation. Consequently the thermal gradient across the encapsulant will be more directly affected by pressure changes. On the other hand, increasing the pressure increases the heat transfer coefficient at the crystal-gas surface. In this case heat extraction from the crystal is improved reducing thermal gradients in the gas. The net effect of increasing the convective heat transfer coefficient at the crystal surface and increasing the gradients across the encapsulant could result in a stronger effect in the stress distribution in the crystal than any changes in the convective heat transfer coefficient. 14 3-144 The experimental results of Emori et al . are in agreement with the effect of pressure on the thermal gradient across the encapsulant discussed above. For argon gas, reducing 2 the pressure from 20 to 5 kg/cm reduces the thermal gradient in the encapsulant from approximately 160°C/cm to 140°C/cm ; the thickness of encapsulant is not given. For neon and helium gases larger gradients were measured, with neon having the largest gradient. Changing gas pressure in neon has a stronger effect on gradients than in helium. They have explained the results by calculating the heat transfer coefficients for the different gases and pressures. For the calculation heat transfer from a vertical wall was assumed. In this case the heat transfer coefficient equation is written as h = 0.53 (k/1) (Gr Pr)1/4 (9.2) 2 33 This equation involves the same quantities as in Equation (9.1). The differences are the values of the numerical coefficients, the exponents affecting the quantities and the meaning of the characteristic length "1". In this case "1" is the length of the crystal which acts as a vertical wall. The calculations show a very good correlation between the gradient in the boron oxide for the different gases and pressures with the heat transfer coefficient values. This correlation is reproduced in Figure 9.43, curve A. The correlation, however, is basically inconsistent since the measurements were made without a crystal being present. Moreover, introducing the crystal and using a larger heat transfer coefficient will tend to reduce the temperature gradient across the encapsulant as explained above. Fortuitously, the correlation in Figure 9.43 (curve A) gives the correct tendency but through a different mechanism. The gas pressure affects the temperature gradient across the encapsulant through changes in the heat transfer coefficient at the encapsulant-gas surface. A correlation between heat transfer coefficient values for heat transfer from an horizontal wall 14 3 14 4 (Equation (9.1)) and measured gradients from Emori et al . ' gives the results shown in Figure 9.43, curve B. In this case a deviation from linearity is observed. The correlation, however, is consistent with the physical process analysed. From the model results and the above discussion it is possible to conclude that the gas pressure and gas composition 234 Vertical Heat Transfer Coefficient (W/cm K) £ u O c cu •o o v. o 5 id3 10 id3 2 00 1 1 1 1 1 1 1 1 1 / s , / ^^^^ 1 1 1 _ A V ^x - <r sY 1 1 1 1 1 1 1 1 1 1 1 0 I 2 Horizontal Heot Transfer Coefficient (Arbitrary Units) Gas Ar N2 He Pre s su re kg/cm^ 20 , 5 20 , 5 20 , 5 Verticol o & O Horizontal Figure 9.43 Correlation between the measured temperature grad ients across the encapsulant and the heat transfer coefficient values as a function of gas pressure and gas nature. Curve A from Refs. 143-144 using convection from a vertical wall ; and curve B using convection from an horizontal surface. 235 may affect the stress distribution in the crystal. A change in the heat transfer coefficient between the crystal and the gas is not enough to explain the large effect on dislocation density changes associated with gas pressure and composition. A full description of the effect of these variables requires the inclusion of the heat transfer mechanisms at the encapsulant-gas interface. A decrease in gas pressure decreases the heat flux to the gas at this surface which results in smaller thermal gradients across the encapsulant. Such an environment will give lower stresses in the crystal. Since the temperature profile in the environment is input in the model and reliable data on the effect of gas pressure and compositon is not available, precise calculations on this matter were not done. 9.7 Curvature of the Solid-Liquid Interface The effect of a non-planar solid-liquid interface on the stresses is studied considering convex and concave interface shapes with respect to the melt. In both cases the interface is assumed to be a paraboloid of revolution which is represented by the following analytical functions £ = ±0.3 p2 p and 5 are the nondimensional radial and axial position respectively with the origin situated at the axis of the crystal ; the + and - signs correspond to the convex and concave interfaces respectively. The value of the constant 0.3 gives the 236 relative displacement between the center and edge of the interface. The mesh used in this case is shown in Figure 9.44(a) for a convex and (b) for a concave interface. They are generated manually. The effect of a number of growth conditions combined with the interface shape is considered. The most important observations follow. 9.7.1 Convex interface The effect of a convex interface combined with the effect of crystal length, thermal conditions and crystal radius is considered. 9.7.1.1 Crystal Length Conditions : R, 27.5 mm ; CA, 30° ; AP, 30 atm. ; B , 21 mm ; Temperature profile as in Figure 8.19, curve 4, CL : (a) 27.5 mm = R, (b) 55.0 mm = 2 R, (c) 82.5 mm = 3 R . The temperature and Von Mises stress fields for a crystal length of 55 mm are shown in Figure 9.45. In the right part of the figure the temperature field is different than that for a crystal with a flat interface. In the lower half of the crystal the isotherms present a large curvature following the interface shape and giving negative radial gradients. In this part of the crystal the axial gradients are larger at the crystal edge than 237 Figure 9.44 Element and nodal configuration at the interface for crystals with curved interface, (a) Convex interface, (b) Concave interface. 238 at the axis. The axial gradients at the axis do not change with distance from the interface. The Von Mises stress field which is shown in the right part of Figure 9.45 is different than the distribution obtained with a planar interface. The largest VMS is observed to be in two regions in the crystal, below the crystal shoulder and at the edge of the solid-liquid interface. The origin and nature of the first stress concentration was discussed previously for the case of a planar interface. The magnitude of the VMS in this region is about 14 * less than for a planar interface. The stress concentration at the interface edge, which is unique for a curved interface, shows stress values which are 30 to 40 * larger than the stress below the shoulder. On the other hand the region with low stresses, which for a planar interface is at the interface edge, has moved to the center and lower part of the crystal near the axis. The lowest stresses are three times smaller than for a planar interface. The largest stress at the interface edge is associated with the large radial and axial gradients in that region. The lowest stress close to the crystal axis and about 2 cm from the interface is associated with the nearly constant gradient in this region. The MRSS contours for the three crystal lengths are shown in Figure 9.46(a-c). Comparing the right part of Figure 9.45 and Figure 9.46(a), it is observed that the VMS and MRSS relative 239 Figure 9.45 Temperature (10 C) and Von Mises stress (MPa) fields for a crystal with a convex interface shape. Radius 27.5 mm. Length 27.5 mm. Encapsulant thickness 21 mm. Temperature profile as shown in Figure 8.19, curve 4. 240 Figure 9.46 MRSS (MPa) contours for a crystal with convex inter face at three lengths, (a) 27.5 mm, (b) 55 mm, (c) 82.5 mm. The largest stresses at the interface edge are (a) 7.22 MPa, (b) 9.19 MPa, (c) 9.66 MPa. Radius 27.5 mm, Boron oxide thickness 21mm. Profile as in Figure 8.19, curve 4. 241 distributions are similar. The absolute levels for the MRSS are approximately half of the corresponding VMS at any point. The effect of crystal length in the MRSS distribution is shown in Figure 9.46(a-c). Comparing the three distributions it is possible to distinguish two regions with specific and reproducible characteristics. One is the region close to the interface which extends to a distance of approximately a radius from the interface at the axis. The second is the crystal above this region. The stress distribution, in the lower part of the crystal, experiences little change with crystal length and is characteristic of the convex interface shape. The region above this has a stress distribution similar to the distribution in a crystal with flat interface. From this it is concluded that the effect of this convex interface does not extends further than a length comparable to the crystal radius. The largest stress value at the interface edge increases with crystal length from (a) to (b) and slightly increases from (b) to (c). This last effect is attributed to a slight increase in axial temperature gradient. The stress distributions of the MRSS-CRSS (Yield) is shown in Figure 9.47(a-c) for the three crystal lengths. In (a) there are two regions with zero stress which are associated with the regions with the lowest stresses from Figure 9.46(a). This zero stress region also appears in (b) and (c). In (c) the upper part of the crystal shows a large area with zero stress. 242 9.47 MRSS-CRSS (Yield) (MPa) contours for a crystal with convex interface at three lengths (a) 27.5 mm, (b) 55 mm, (c) 82.5 mm. Radius 27.5 mm, boron oxide thickness 21 mm. Profile as in Figure 8.19, curve 4. 243 The dislocation density in the crystal at the end of growth depends on the position in the crystal. The large MRSS-CRSS (Yield) in Figure 9.47(a-c) at the interface edge suggests that the dislocations at the crystal surface are formed right after solidification and not above the encapsulant surface as with the plane interface. The axial MRSS-CRSS (Yield) near the crystal surface is lowest close to the cone, gradually increasing with distance from the cone. At a distance of 50 mm from the cone the MRSS-CRSS (Yield) is double that close to the cone. At the centre, the largest stresses at a distance of a radius from the interface is half the stress at the edge. At the seed and bottom ends of the crystal the stresses are low and never reach the maximum stresses. At the seed end, the maximum stress is always less than 3.0 MPa that is, half the value at the edge. Similar results are obtained at the tail end where the maximum stress at the centre occurs very close to the interface. Thus cross-sections at any distance from the cone will have a W-shaped MRSS-CRSS (Yield) distribution along a diameter except at the tail end where the MRSS-CRSS (Yield) distribution is U-shaped. The edge of the U is three times higher than the centre. The symmetry of the MRSS-CRSS (Yield) in (001) planes depends on the position of the plane with respect to the cone. Close to the cone the MRSS-CRSS (Yield) has the symmetry shown in Figure 9.48. This cross-section is at a distance of 11.0 mm from the cone when the crystal is 27.5 mm long (Figure 9.47(a) ). The maximum stress is reached at this point. 244 Figure 9.48 MRSS-CRSS (Yield) (MPa) contours in a (001) plane at a distance of 11 mm from the cone in the crystal shown in Figure 9.47(a). 245 The symmetry of the MRSS-CRSS (Yield) in sections which are more than 55 mm far from the ends of the crystal at the end of growth will be determined by the sequence of symmetry changes shown in Figure 9.49(a-c). Despite the fact that these sections were taken from only one crystal (Figure 9.47(c)), it was observed that all the sections far from the ends pass through the same symmetry patterns and stress levels. Close to the interface (part (a) ) there is four-fold symmetry with an U-distribution along a diameter. At the edge eight-fold symmetry is present. In this region the dislocations are expected to have the same symmetry. Sections from the tail end show a symmetry of the dislocations as shown in Figure 9.49(a). 9.7.1.2 Effect of Encapsulant Thickness and Gradient Conditions : B, 50 mm ; Temperature profiles as in Figure 9.35(b), The results for the stress distributions for two gradients are shown in Figure 9.50(a-b) for a 50°C/cm gradient and Figure 9.51(a-b) for a 12.5°C/cm gradient. Part (a) in each figure corresponds to the MRSS distribution and part (b) to the MRSS-CRSS (Yield) distribution. The MRSS distributions for the two gradients show a concentration of stress at the interface edge and below the shoulder. The stress at the interface edge is about 80 % larger than below the shoulder. The lowest stress level occurs at the 246 a 247 248 Figure 9.49 MRSS-CRSS (Yield) (MPa) contours in (001) planes at three distances form the cone. (a) 79.75 mm, (b) 66.0 mm, (c) 52.25 mm. Section corresponds to the crystal shown in Figure 9.47(c). 249 Figure 9.50 (a) MRSS (MP) and (b) MRSS-CRSS (Yield) (MPa) for a crystal with convex interface. Boron oxide thickness 50 mm. Radius 27.5 mm. Length 55 mm. Average gradient 50°c/cm. 250 Figure 9.51 (a) MRSS (MP) and (b) MRSS-CRSS (Yield) (MPa) for a crystal with convex interface. Boron oxide thickness 50 mm. Radius 27.5 mm. ' Length 55 mm. Average gradient 17°C/cm. 251 centre of the crystal. The largest stress value at the interface edge decreases proportionally with a reduction in gradient. The MRSS-CRSS (Yield) in part (b) of the figures for the two gradients shows zero stress regions. The area of this region substantially increases with the gradient. For both conditions large areas close to the interface edge show stresses which are larger than the CRSS (Yield). 9.7.1.3 Effect of crystal Radius Conditions : R, 40 mm ; CL, 80 mm ; B, 21 mm ; Temperature profile a in Figure 8.19, curve 4. The MRSS and the MRSS-CRSS (Yield) for this crystal are shown in Figure 9.52(a) and (b) respectively. The stress distributions are similar to those obtained with the 27.5 mm crystal with two common characteristics. Large stress levels exist at the interface edge and above the encapsulant surface. The largest stress occurs at the interface edge. It is important to note that while the stress level above the encapsulant surface increases as the radius increases (45 %), the stress level at the interface edge only increases about 10 %. On the other hand the symmetry of stresses in (001) planes show the same features as described for the 27.5 mm crystal. 252 253 9.7.2 Concave Interface Conditions B, 21 mm Temperature profile as in Figure 8.19, curve 4, a) R, 27.5 mm CL, 55 mm b) R, 40 mm CL, 80 mm. The results for the MRSS-CRSS (Yield) for the two crystal radii are shown in Figure 9.53(a) and (b). Part (a) for the 27.5 mm radius and part (b) for the 40 mm radius. As in the case of the convex interface, the stress distribution may be divided in two regions. One close to the interface and extending a distance from the interface comparable to the radius and another formed by the region above this section. The low region has stress distributions which are similar for both crystal radii and are characteristic of the concave interface shape. Very large stresses are present at the interface with largest stress at the centre of the crystal. In the second region, well above the interface, the stress distribution is similar to that obtained with a planar interface. The largest stresses occur at the interface and therefore the dislocation density is controlled by the stress distribution at the concave interface. For this interface shape, the stress distribution in the radial direction follows the W shape with the centre about 15 * higher than the edges. The effect of increasing the radius for a concave interface is different than that obtained for a convex interface. For 254 Figure 9.53 (a) MRSS (MP) and (b) MRSS-CRSS (Yield) (MPa) for a crystal with concave interface, (a) Radius 27.5 mm, (b) 40 mm. Boron oxide thickness 21 mm. 255 the concave interface increasing the radius about 45 % increases the largest stress value from 14.26 MPa to 18.86 MPa (30 *). The symmetry of the dislocations formed close to the interface is shown in Figure 9.54. This (001) plane corresponds to a section at 2.75 mm from the interface for the 27.5 mm crystal. Similar symmetry is observed for the 40 mm crystal. In the central region eight-fold symmetry is observed. Four fold symmetry is also observed in a ring close to the edge. In this case typical eight-fold at the edge is not observed. 256 Figure 9.54 MRSS-CRSS (Yield) (MPa) contours in a (001) plane at a distance of 2.75 mm from the interface in the crystal shown in Figure 9.53(a). 257 CHAPTER 10 RESULTS FOR COOLING AFTER GROWTH Using the combined analytical and numerical scheme described in Chapter 7 and evaluated in Chapter 8 the temperature and stress fields during cooling after growth are calculated. Results are obtained for a crystal with a radius 27.5 mm and a length of 110.0 mm. This length corresponds to the largest length for which stress and temperature calculations are made during growth. The axial temperature at the end of growth is the initial temperature field for the cooling model with the radial gradient neglected. In the calculation of the thermal and stress fields in the crystal after solidification, the initial temperature profile in the crystal, and the thermal characteristics of the media surrounding the crystal are of major significance. In the calculations, three temperature gradients in the region of length R above the interface are considered, 70, 35 and 17.5°C/cm designated as GC , GC/2 and GC/4. These gradients are about half the gradients in the growing chamber during solidification. The crystal as it cools, is surrounded by either argon at 30 atmospheres or boron oxide. The argon temperatures considered are 800 and 1000°C and the boron oxide temperature is taken as 1000°C . 258 10.1 Ambient Temperature 1000°C 10.1.1. Argon 10.1.1.1. Initial Gradient GC = 70°C/cm The temperature distribution over the right half of a vertical section of a crystal after 5 seconds from solidification is shown in Figure 10.1(a). The results show that in a region of about 1 cm adjacent to the interface, the temperature has dropped rapidly. The axial gradient in this region changes from positive to negative. The radial gradients are high, and the temperature drop at the edge is large, about 80°C. The radial gradient is negative between the interface and the 1000°C isotherm, and then becomes positive at lower temperature. With a changing ambient temperature, which is normally the case, ambient temperatures will be lower than the crystal surface temperatures resulting in negative radial gradients throughout the crystal. The assumption of constant ambient temperature in the model will likely give calculated stress levels which are too low. This error will become less significant as the ambient temperature gradients become smal1er. The temperature distribution 10 seconds after solidification is shown in Figure 10.1(b). Comparing (a) and (b) the curvature of the isotherms near the interface has increased giving higher radial gradients. In (b) the axial gradient at the interface is less than in (a) and the distance above the interface where the axial gradient changes sign has increased. 260 c d Figure 10.1 Temperature (103oC) field during cooling at four times. (a) 5 s, (b) 10 s, (c) 20 sfa (d) 60 s, Initial gradient close to interface 70 C/cm. Argon temperature 1000 C 261 The temperature distribution after 20 seconds (Figure 10.1(c)) shows continuing high radial gradients close to the interface and a considerably decreasing axial gradient. After 60 seconds (Figure 10.1(d)) the temperature distribution has become much more uniform. By 300 seconds, the calculations show the temperature distribution is almost uniform throughout the crystal. The very rapid cooling in the first five minutes after solidification supports the quasi-steady state approximation assumed in the calculations of the temperature distribution during crystal growth. Further examination of the isotherms in Figure 10.1(a-d) shows that mixed control of the heat transfer process occurs during cooling. This can be concluded from the observation that the 1000°C isotherm does not move considerably from a to d in Figure 10.1. Heat transfer control at the surface would considerably move such isotherm upwards in the crystal. On the other hand, heat transfer control in the bulk crystal would give larger than the observed radial gradient. The MRSS-CRSS (Yield) determined from the temperature distributions shown in Figure lO.l(a-d) are shown in Figure 10.2(a-d). The largest stresses are observed to occur at the interface. A low stress region is present about 1 cm above the interface midway between the outside surface and the vertical axis which moves towards the axis with time. As a result of the 263 c d Figure 10.2 MRSS-CRSS (Yield ) (MPa) during cooling for the four temperature fields shown in Figure lO.l(a-d). (a) 5 s, (b) 10 s, (c) 20 s, (d) 60 s. Initial gradient 70°C/cm. Argon temperature 1000 C. 264 movement the stress distribution along the crystal diameter changes from a W to a U shape with time. The largest MRSS-CRSS (Yield) near the interface occurs about 10 seconds after growth stops, and ranges between 18 and 4.5 MPa. These stress values are appreciably larger than the values at the same position in the crystal during growth and are about 3 times the stress in the crystal above the boron oxide when growth is stopped. This would result in a three-fold increase in dislocation density in the vicinity of the end of the crystal generated during cooling. The MRSS-CRSS (Yield) for crystal cross-sections at 2.75, 8.25, 16.5 and 27.5 mm from the interface are shown in Figure 10.3(a-d), the main characteristics of the stress symmetries observed are : 1) At 2.75 mm from the interface the outer ring has an eight-fold symmetry and the stress distribution across a diameter is U shaped. 2) At 8.25 mm from the interface the centre has four-fold symmetry and two-fold symmetry.. 3) At 16.5 mm from the interface there is circular symmetry. There are four minimum stress regions in the <100> directions. 4) At 27.5 mm from the interface there is two-fold symmetry. 265 266 267 C 268 Figure 10.3 MRSS-CRSS (Yield ) (MPa) contours in (001) planes at four distances from the bottom (interface) in the crystal shown in Figure 10.2(b) 10 s, (a) 2.75 mm. (b) 8.25 mm (c) 16.5 mm (d) 27.5 mm . 269 The stress distribution and symmetry are time dependent. At 2.75 mm from the interface the eight-fold symmetry in Figure 10.3(a) at 10 seconds changes at 60 seconds to the eight-fold symmetry shown in Figure 10.4(a). At 8.25 mm from the interface, the two and four-fold symmetries in Figure 10.3(b) at 10 seconds change at 60 seconds to the two-fold symmetry shown in Figure 10.4(b). 10.1.1.2 Initial Gradient GC/2 = 35°C/cm In this case the stresses reach a maximum at 10 seconds. The temperature field is shown in Figure 10.5(a) and the corresponding MRSS-CRSS (Yield) field in Figure 10.5(b). Comparing Figure 10.5(a) with Figure 10.4(a) for a 70°C/cm gradient it is noted that the isotherms for the 35°C/cm gradient have larger curvature than for the 70°C/cm gradient. The MRSS-CRSS (Yield) distribution in Figure 10.5(b) is similar to the distribution in Figure 10.4(b) for the 70°C/cm gradient. Large curvature in the isotherms and large stresses are still present at 60 seconds as shown in Figure 10.6(a-b). The MRSS-CRSS (Yield) in a cross-section at 2.75 mm from the interface has eight-fold symmetry. At 8.25 mm the stress has a strong two-fold symmetry (Figure 10.7(b)). The two-fold symmetry gradually changes to a near circular symmetry with small traces of four-fold symmetry given by weak local minima in the <110> directions as shown in Figure 10.7(b). The U-shaped stress distributions along a diameter is present in all cross-sections. 270 271 Figure 10.4 MRSS-CRSS (Yield ) (MPa) in (001) planes at two distances from the bottom in the crystal shown in Figure 10.2 (d) 60 s, (a) 2.75 mm (b) 8.25 mm. 272 b Figure 10.5 (a) Temperature field (10 C) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in argon at 1000 C after 10 s. Initial gradient 35°C/cm. 273 Figure 10.6 (a) Temperature field (10°°C) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in argon at 1000 C after 60 s. Initial gradient 35°C/cm. 274 a 275 Figure 10.7 MRSS-CRSS (Yield ) (MPa) in (001) planes at two distances from the bottom in the crystal shown in Figure 10.5 (b) 10 s. (a) 8.25 mm (b) 27.5 mm. 276 10.1.1.3 Initial Gradient GC/4 = 17.5°c/cm The temperature field (Figure 10.8(a)) gives isotherms which have the highest curvature among the initial gradients considered. The MRSS-CRSS (Yield) (Figure 10.8(b) field in the region close to the interface is similar to the previous cases. The region with large stress near the crystal surface and adjacent to the interface is expanded to encompass nearly all of the crystal. The stress distribution along the diameter is U-shaped. The MRSS-CRSS (Yield) in cross-sections shows 1) eight-fold symmetry close to the interface 2) two-fold symmetry above the interface and up to a distance from the interface of 16.5 mm 3) Near circular symmetry at 27.5 mm from the interface with low stresses in the centre covering more than half the section surface area and large stress gradients at the edge. These give a U-shaped stress distribution with a broad base. The large stresses persist after 60 seconds but they are not larger than the ones observed at 10 seconds. The stress distribution in cross-sections at 60 seconds is similar to the distribution at 10 seconds. Along a diameter the stress distribution tends to take a V shape rather than U shape. At a distance of 27.5 mm from the interface the central region in the 277 Figure 10.8 (a) Temperature (Yield ) (MPa) at 1000 C after field (10OUC) and (b) MRSS-CRSS field for a crystal cooling in argon 10 s. Initial gradient 17.5°C/cm. 278 wafer shows a well distinguished eight-fold symmetry (Figure 10.9). The stress level in this section is higher than at 10 seconds and therefore the symmetry may be evident in the dislocation density distribution. 10.1.2. Boron Oxide The calculations are repeated for the same initial thermal fields in the crystal used above 70°C/cm and 17.5°C/cm and assuming the surrounding medium is boron oxide. The temperature and stress fields for the three thermal gradients GC, GC/2 and GC/4 are shown in Figure 10.10 to 10.12. Parts a and b show the two fields in that order. Results at 10 seconds are shown which exhibits the highest cooling stresses. The MRSS-CRSS (Yield) distributions in Figure 10.10(b) to 10.12(b) are similar in shape to the distributions obtained with an argon atmosphere, but stresses are 20 * higher in magnitude. The MRSS-CRSS (Yield) in cross-sections present some distinctive characteristics. At 10 seconds and 27.5 mm from the interface the two-fold symmetry is replaced by a near circular symmetry which is also present at 16.5 mm. In addition cross-sections at 5 seconds show that the eight-fold symmetry close to the interface and the two-fold symmetry above do not form instantly but evolve from a close to circular symmetry and a four-fold symmetry respectively. 279 Figure 10.9 MRSS-CRSS (Yield) (MPa) in (001) planes a distance of 27.5 mm from the bottom in the crystal shown in Figure 10.8 (d) 10 s. 0 280 a b Figure 10.10 (a) Temperature field (10 C) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in boron oxide at 1000°C after 10 s. Initial gradient 70 C/cm. 281 Figure 10.11 (a) Temperature field (10°"c) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in boron oxide at 1000°C after 10 s. Initial gradient 35°C/cm. 282 b Figure 10.12 (a) Temperature field (10 C) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in boron oxide at 1000°C after 10 s. Initial gradient 17.5°C/cm. 283 10.2 Ambient Temperature 8QQ°C/CIB In this case only an argon atmosphere is considered since the boron oxide layer is generally well above 800°C. The temperature and stress fields after 10 seconds for three initial temperature gradients are shown in Figures 10.13 to 10.15. The stress distributions are seen to be similar to that obtained for 1000°C, and the stress levels between those obtained when the surrounding media was argon and boron oxide. The stress fields in the cross-sections exhibit a few different features, the most prominent difference being the lack of local minima in the<100>directions. 10.3 Analysis of the Results for Cooling The most significant result from the results described in the last section is that all conditions examined gave much larger stresses than the stress levels developed during growth. This clearly indicates the importance of proper cooling to minimize dislocation density in the tail end of the crystal and in some cases in the entire crystal. Cooling in argon is shown to produce smaller stresses than cooling in boron oxide. This result, in a very broad sense, can be directly correlated with the heat transfer coefficient, since larger heat transfer coefficients produce larger stresses. The effect of the heat transfer coefficient is such that a larger value will cool the surface more efficiently, leading to larger thermal gradients in both radial and axial directions. 284 Figure 10.13 (a) Temperature field (10 C) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in argon at 800°C after 10 s. Initial gradient 70 C/cm. 285 Figure 10.14 (a) Temperature field (lO^c) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in argon at 800°C after 10 s. Initial gradient 35°C/cm. 286 Figure 10.15 (a) Temperature field (10OUC) and (b) MRSS-CRSS (Yield ) (MPa) field for a crystal cooling in argon at 800°C after 10 s. Initial gradient 17.5 C/cm. 287 In addition to the external conditions, the conditions in the crystal at the start of cooling must also be considered. Low thermal gradients in the crystal gives stresses along the surface of the crystal as a result of the large radial gradients. However, the magnitude of the stresses near the interface cannot be explained in terms of radial and axial gradients separately. This is shown in Figure 10.16 for a 800°C argon temperature and Figure 10.17 for a 1000°C boron oxide temperature. In each case two initial gradients are considered, (a) GC and (b) GC/2. In Figure 10.16 the comparable stresses in both cases is attributed to the larger radial gradient in curve (a) and larger axial gradient in curve (b). In Figure 10.17 the larger radial gradient in curve (b) than in Figure 10.16 can account for the larger stresses. The largest stress for initial condition GC (a) in Figure 10.17 cannot be accounted for by a difference in axial or radial gradient. In this case the largest stress is attributed to the rapid change in radial gradient (curvature) with distance from the interface observed. The changes in radial gradient create more constrains for a free thermal expansion which leads to higher stresses. The above analysis shows the advantage of calculating thermal stresses using the axisymmetric approximation over the plane strain approximation which cannot account for axial temperature variations. 200 " 1120-1080 -1040 -Figure 10.16 Temperature profiles in the crystal cooling in argon at 800 C for two initial gradients in the crystal, (a) 70 C/cm (b) 35 C/cm. The right part corresponds to the radial profiles and the left part to the axial profile. Full and broken lines correspond to the axis and surface temperatures respectively. Figure 10.17 Temperature profiles in the crystal cooling in boron oxide at 1000°C for two initial gradients in the crystal. (a) 70°C/cm (b) 35 C/cm. The right part corresponds to the radial profiles and the left part to the axial profile. Full and broken lines correspond to the axis and surface temperatures respectively. to oo 290 CHAPTER 11 SUMMARY AND CONCLUSIONS In the present Investigation the stress distribution in GaAs crystals has been calculated, analysed and related to dislocation formation during solidification and cooling. The calculations were made assuming that the material is thermoelastic, and that axisymmetric conditions apply. The numerical solutions of the differential equations were obtained employing finite element techniques. For the temperature calculations the quasi-steady state heat conduction equation was employed and the element equations were obtained using the Galerkin's method. The boundary conditions assumed constant temperature at the solid/liquid interface and Newton's law of cooling at the crystal surface. Linear triangular toroidal elements were used to discretize the crystal. The element equations for the stress calculations were obtained using the minimum energy principle. Linear and quasi-quadratic elements were used to discretize the crystal domain. Numerical values of the finite element solutions satisfying simplified boundary conditions were compared with numerical values of analytical solutions. In the stress calculations, numerical solutions were obtained for radial and axisymmetric temperature fields and compared with plane strain and axisymmetric analytical solutions. In addition, the temperatures calculated with the numerical method were compared with 291 temperatures measured in a high pressure grower. The mathematical model for the temperature and stress fields were evaluated on the basis of these results. The most important results and conclusions obtained from the evaluation are given below. Temperature Field 1. The finite element solutions of the temperature equation gave results in good agreement with the numerical values of the analytical solutions, obtained for the same problem. For the analytical solutions simplified boundary conditions were used consisting of constant temperatures at the end of the crystal and constant heat transfer coefficient at the crystal-gas interface. The heat transfer coefficients used were 0.3 and 0.6 cm The crystal was taken as a cylinder of aspect ratio two . 2. Refinement of the mesh in the finite element solutions did not change the numerical values of the calculated temperatures. 3. The calculated temperatures are in good agreement with reported temperature measurements in GaAs crystals in a Melbourn crystal grower at 3.04 MPa pressure. 292 Stress Field 4. The stress components calculated by the finite element method converge to different values depending on the initial strain values (nodal temperatures) used in the calculations. 5. For linear and quasi-quadratic displacements inside the element, a linear initial strain in the element gives stress components close to one order of magnitude higher than a constant initial strain. 6. For a given nodal configuration, the linear element formul ation with a constant initial strain gives stress components that converge faster than with the pseudo-quadratic element. This is attributed to the use of an incomplete B matrix in the pseudo- quadratic element. 7. For radial temperature fields, the stresses calculated by finite element using average temperatures are in good agreement with analytical axisymmetric solutions. They agree less well with analytical plane strain solutions. The finite element solutions using linear initial strains differ markedly from the anlytical solutions. 8. For axisymmetric temperature fields the stresses calculated with the finite element method are three times smaller than the stresses calculated using the plane strain approximation. The analytical axisymmetric stress values agree less well with the finite element solutions due to the divergency of 293 the Bessel function of first order for large numbers of terms in the Fourier series employed in the solutions. Specific results on the model formulation were obtained. Calculations were made for a crystal length of 10 mm, radius 20 mm, cone angle 45°, encapsulant thickness 10 mm. The results consist of the following. 9. Calculated stress values have been reported in which the 12 RSS values are added to give a local RSS. Adding the 12 RSS gives stress distributions with stress levels which are 7 times larger than the MRSS and 3 times larger than the VMS distributions. Adding the RSS components is inconsistent with the concept of a tensorial quantity. 10. Changing the heat transfer coefficient in a span representing 53 % of the original values does not substantially change the temperature and stress distribution in the crystal. The stress levels change only 20 % 11. Changing the temperature gradient in the environment surrounding the crystal has a strong effect on temperature and stress distributions in the crystal. The axial temperature gradient in the crystal is usually half the gradient of the environment, except at the highest gradient of 400°C/cm used which gives axial gradients one quarter of the environment gradient. The maximum stress in the crystal decreases proportionally to the decrease in gradient of the environment except at the lowest gradients near 50°c/cm. 294 12. A stress concentration usually occurs below the crystal shoulder for crystal lengths comparable to the encapsulant thickness. The concentration disappears at low gradients (50°C/cm). 13. Increasing the gradient from 100°C/cm to 400°C/cm has little effect on the operative slip system. The changes on stress distribution symmetries in transverse planes are minimal. 14. The stress field on transverse sections of the crystal usually exhibits eight-fold symmetry at the edge. The symmetry is associated with the slip mode of the MRSS which has an eight-fold distribution. 15. Increasing the encapsulant thickness from 10 mm to 25 mm without changing the temperature profile in the environment surrounding the crystal has little effect on the temperature and stress distribution. 16. The temperature profile shape in the boron oxide significant ly influences the stress level and stress distribution in the crystal. a) A constant gradient across the encapsulant gives the lowest stress values. b) A non-constant gradient in the boron oxide changes the stress symmetry, resulting in stronger four-fold symmetry on transverse planes. 295 The model was used to study the effect of the growth condition. The main results and conclusions are given below. Cone Angle The cone angles considered were 7°, 30°, 45°, 54.7° and 65°. Encapsulant thickness and crystal length were 10 mm crystal radius 20 mm. The results indicate that : 17. The VMS and MRSS distributions are similar for each cone angle. 18. The VMS exhibits a minimum stress level for the cone angle 54.7° which corresponds to a cone parallel to the (111) plane. At low cone angles the stress behaviour can be accounted for by the decrease in radial temperature gradients. At large cone angles the stress increases possibly due to constraints in the strain field. 19. The MRSS levels are independent of the cone angle. 20. The MRSS gives levels generally half the VMS levels. At 54.7° cone angle the MRSS is two times larger than the VMS. The singularity is due to the condition of no traction at the cone surface. For a cone surface coinciding with a (111) plane the RSS components are at a maximum. 21. The stress symmetry in transverse planes changes with cone angle. At 2.5 mm from the interface the symmetry in the central area changes from eight-fold for 7° cone angle to 296 four-fold for larger angles. At transverse sections 50 mm from the interface the symmetry changes from four-fold for the 7° cone angle to axisymmetric for the larger angles. At the edge of the sections there is always eight-fold symmetry. For the given growth conditions these symmetries may not be apparent in dislocations distributions because the stress levels are lower than the critical yield stresses. Crystal Length The crystal lengths considered were up to 110 mm long. Two crystal radii were studied 27.5 mm and 40.0 mm. The conclusions are valid for a 21 mm encapsulant unless it is specified that the same conclusion is valid for other conditions. 22. For any crystal length the radial stress distribution is W-shaped with higher stresses at the crystal centre and near the vertical surfaces. The valleys in the W tend to be closer to the outside surfaces near the top and bottom ends of the crystal. & 23. Lower stresses are observed at both ends of the crystal for crystal lengths larger than the encapsulant thickness. 24. For a crystal shorter than the encapsulant thickness, there is a stress concentration below the. shoulder. For longer crystals there is a stress concentration above the encapsulant surface. These are valid for any encapsulant thickness and temperature gradients in the environment 297 surrounding the crystal. It should be noted that this may not be the case for short crystals and linear temperature profiles in the encapsulant as described in paragraph 12 above . 25. The stress level increases with crystal length reaching a maximum for a crystal length twice the encapsulant thickness. The stress slightly decreases for longer crystals. This result is valid for all encapsulant thicknesses and thermal gradients studied. The maximum stress above the encapsulant is due to the thermal discontinuity at the encapsulant surface. When this coincides with the mid-length position in the crystal quasi-plane strain conditions contribute to give the largest stress level during growth. 26. The slip mode corresponding to the MRSS cannot be associated with a stress level. In longitudinal (010) planes the most common modes are III, IV and V. 27. The stress symmetry in transversal (001) planes shows the following patterns : a) There is always eight-fold symmetry at the edge of the sect ions. b) The four-fold symmetry is stronger closer to the solid/ liquid interface. It also appears at the seed end. 298 c) Two-fold symmetry results from a degeneration of the four-fold symmetry. It is manifested by an elongation of the minima in the <110> and <110> directions. d) Far from the ends, there is axisymmetry, except at the edge of the section where there is eight-fold symmetry. e) For crystal lengths larger than two radii and sections far from the interface, the minimum stress levels are in the <100> directions rather than in the <110> directions. 28. The dislocation distribution predicted with the model for normal growth conditions shows that : a) The dislocation density at the seed and tail ends are lower than far from the ends. b) The four-fold symmetry always forms close to the inter face. c) There is a correlation of stress symmetry with distance from the cone between two crystals with radius 27.5 mm and 40.0 mm. Growth Velocity _ 3 29. For growth velocities of 10 cm/sec and lower, the temper ature field is at steady state, and the effect of growth velocity on the stress field is negligible. For larger velocities larger gradients and stresses are obtained. 299 Effect of Radius 30. Increasing the radius increases the stress levels. There is little change on the relative stress distribution. The effect of increasing the radius on stress levels depends on the thermal conditions in the environment. For thin encapsulants (21 mm) and normal gradients (100°C/cm) the relative stress level increment is similar to the relative radius increment. For thick encapsulants (40-50 mm) and/or low thermal gradients the relative effect of increasing the radius is 2 to 3 times larger than the relative radius increment. Effect of Thermal Conditions Increasing the encapsulant thickness and decreasing the thermal gradient in the environment surrounding the crystal decreases the stress level. The relation between stress level and gradient reduction is nearly linear. The radial and axial temperature gradients in the crystal do not significantly change with a change in crystal radius from 27.5 mm to 40.0 mm. The stress level change cannot be accounted for by changes in the thermal gradients. The higher stress level with the larger radius is due to an incompatibility of the strain field in a cylinder arising from radial temperature gradients. A given radial gradient creates larger stresses in a larger crystal. 31 . 32 . 33 . 300 34. To obtain large areas (80 * of the crystal) in which the stresses are lower than the yield stress, an average gradient in the surrounding environment should be less than 10°C/cra for the 27.5 mm crystal and 7°C/cm for the 40.0 mm crystal. This requires that the gradient during normal growth be reduced by a factor of 6 to 8 for encapsulant thickness of 21 mm. Gas Pressure and Nature 35. The large effect of pressure on dislocation density observed experimentally cannot be accounted for by the corresponding pressure dependence of the heat transfer coefficient between the crystal and the surrounding gas. The effect of pressure is likely related to effects in the surrounding gas. The transfer efficiency of the gas will change which can significantly modify the thermal gradients across the encapsulant and in the gas. This is indicated by the larger heat transfer coefficients between the encapsulant surface and gas for higher gas pressure. Solid/Liquid Interface Shape-Convex Curvature 36. A convex interface significantly modifies the stress distrib ution in a region approximately one radius from the interface. The modifications consist of the following a) A stress concentration at the interface edge. 301 b) A U-shaped stress distribution close to the interface. c) The above characteristics do not change with crystal length. d) Highest stress values are obtained at the edge of the interface for all crystal lengths. This indicates that dislocations are generated immediately after solidification. e) Stress levels are reduced by increasing the encapsulant thickness and decreasing the temperature gradients. The stress concentration is not eliminated by increasing the encapsulant thickness. f) Increasing the crystal radius does not change the stress level or distribution close to the interface. Concave Curvature A concave interface shape substantially modifies the stress distribution ' in regions adjacent to and about one radius thickness from the interface. The modifications consist of the following a) Stress concentrations are developed at the centre and edge of the crystal at the interface. The stress levels are the largest and about twice as large as stress levels observed above the encapsulant surface. 302 b) Increasing the crystal radius does not change the stress distribution. The stress level increases proportionally to the relative change in radius. Cooling the Crystal to Ambient Temperature after Growth The effect on temperature and stress distributions during cooling has been analysed. The temperature fields were obtained using analytical time dependent solutions of the heat conduction equation. Solutions were obtained assuming a parabolic initial temperature profile in the axial direction. It was also assumed that the crystal is immersed in a medium which is either argon or boron oxide and the medium is at a constant temperature. The temperatures were 800°C or 1000°C for the argon atmosphere and 1000°C for the boron oxide. The analytical series solutions were found to converge to 96 % of the theoretical limit value. The stress fields were calculated using the numerical finite element method employed for the stress calculations during growth. The analysis focused on a region adjacent to the interface and extending to a distance of a radius from the interface. The crystal radius and length were 27.5 mm and 110 mm respectively. Temperature profiles showing three gradients were considered. The main results and conclusions follow. 38. The temperature and stress fields during cooling differ markedly from the fields obtained during growth. In the region of interest close to the interface the differences are : 303 a) Larger radial gradients. b) Larger stress levels of the order of two to four times higher than during growth. The specific characteristics of the fields are : a) The largest stress is reached between 10 and 20 seconds after cooling starts. b) In most cases the equilibrium temperature is reached after 5 to 10 minutes. c) The stress level is not strongly dependent on cooling and initial conditions. Larger stresses are obtained with larger radial and axial gradients. These combined with different axial gradients along the axis and at the crystal surface give the largest stresses. d) The stress level decreases rapidly with distance from the interface ; the larger the initial axial gradient is, the shorter the distance. In actual cooling conditions this may not be the case because the temperature in the media is not constant, as assumed. The symmetry of the stress in (001) transverse planes markedly differs from the symmetries obtained during growth. The most important characteristics are : a) The radial stress distribution always shows a U or V shape instead of the usual W shape. 304 b) At 2.75 mm from the interface the eight-fold symmetry is stronger and appears in a larger ring at the edge which is about half the wafer area. c) At 8.25 mm from the interface two-fold symmetry is observed which is formed by an elongation of the two minima in the <110> and <110> directions and a displace ment of these minima to a position closer to the edge of the wafer. d) For distances larger than 8.25 mm the symmetry depends on cooling conditions and time. The differnt symmetries observed are two-fold, eight-fold or nearly axisymmetric with four minima either in the <100> or <110> directions. e) The symmetry is time dependent. The two-fold symmetry evolves from a four-fold symmetry present at an early stage during cooling. The eight-fold symmetry close to the tail end evolves from a nearly circular symmetry. 40. The results suggest that a considerable number of disloc ations may be generated during cooling giving dislocation densities two to four times larger than far from the tail. In this case the larger densities observed in LEC crystals may be accounted for by the stresses generated during the cooling process. The two-fold symmetries observed experimentally in wafers from the seed end support this conclusion. 41. The similarity between the EL2 distributions in wafer from the tail end showing the two-fold symmetry and sometimes 305 minima in the <100> directions supports the mechanism 8 0 81 proposed by Holmes et al. ' for the EL 2 formation. This mechanism as described in Appendix I, involves dislocation climb during cooling. Summary of the Conclusions A mathematical model for temperature and stress calculations during growth and cooling of LEC GaAs has been developed and validated. 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Jaeger, "Conduction of Heat in Solids" 2nd ed., Oxford Press, Oxford, 1958. 320 APPENDIX I EFFECT OF DISLOCATIONS ON GaAs AND DEVICES I.1 Effect of Dislocation on Properties The semi-insulating property of GaAs makes it suitable for use as substrate for the fabrication of integrated circuits, simplifying device isolation and permitting low capacitance 49 interconnections through the substrate . The choice of ion implantation for device fabrication imposes rigorous conditions on substrate properties, especially homogeneity. The semi-insulating property is essential. Inhomogeneous FET performance has been correlated with dislocation distribution in Cr doped and 5 0-53 undoped semi-insulating GaAs . Moreover, on a micro-scale level, it is found that the FET threshold voltage is affected by dislocations located at less than 20-30 microns from the FET. There has been no consensus on how or how much, the threshold 5 4 5 5 voltage is affected ' . Recently, this correlation has been examined more carefully and related to the interaction of dislocations as sources of sinks , or point defects and the EL2 level concentration distribution which is strongly correlated with FET characteristics. In additions, a great amount of effort has been directed to evaluate the role of dislocations on semi-insu1 ating properties. Dislocation distributions have been directly correlated with ii 4. «- -K <-• 56,57 . ... 58-61 leakage current distribution , resistivity , and carrier 321 concentration in Si implanted LEC-growth, semi-insu1 ating 6 2 6 4 GaAs ' . Leakage current and sheet resistance have M-shaped distribution, while resistivity and carrier concentration have the W-shape. In addition, it is found that dislocations affect the sheet carrier concentration within about a 75 micron radius 64 area In Cr-doped LEC GaAs the reports are conflicting. On one hand, it is found that Cr follows a M-shaped distribution inverse of the W-shaped EPD distribution, suggesting that the dislocations can affect the Cr concentration and annealing behaviour of donor impurities^. On the other hand, no variation 6 6 of Cr concentration associated with dislocation is found , and the resistivity variations follow the W-shaped distribution. Moreover resistivity does not correlate with Cr except at high concentrations. It was also reported that Cr concentration across a wafer follows a U-shaped distribution, while the Cr++ 6 7 concentration follows a W-shaped distribution Cr is usually added to the melt to obtain semi-insu1 ating GaAs. At present, extensive studies are being carried out in order to understand the mechanisms that determine the semi-insu1 ating 6 8 properties of GaAs . The production of high resistivity materials depends on a critical compensation among the different dopants present. In the four-level model an excess of shallow donors is compensated by an excess of deep acceptors over deep 6 9 donors . In the three-level model an excess of acceptor over 7 0 donors is compensated by the deep donors . The shallow donors 322 are believed to be due to S and Si or to the intentional doping of Te. Shallow acceptors are due mainly to C, Mn or other impurities. The energy levels of shallow donors and acceptors are ubicated close to the conduction and valence band respectively. The deep acceptor is due to Cr and the deep donor is the so called EL2 or EL2 family. Both are mid-gap levels. For Cr-doped GaAs the four level compensation mechanism applies while for undoped GaAs the three level mechanism applies. The most important commercially available SI substrate is 7 1 Cr-doped. However, the large diffusion coefficient of Cr has detrimental effects in the IC's technology. Attempts to 7 2 substitute Cr for the one order of magnitude slower Vanadium is unlikely because Vanadium plays no direct role in the 7 3 compensation process in GaAs . These two facts lead to concentrate the efforts on undoped SI GaAs and consequently on understanding the nat ure and role of EL2. In this review a short summary of the findings on EL2 is given below because of the interaction of the EL2 with dislocations. 7 3 Originally the EL2 level was associated with oxygen until 7 4 7 6 it was demonstrated that it was not due to this element ' In 7 6 addition Martin et al have shown that it is not related to Cr. Alternatively, a strong correlation with the dislocation distribution is found ; the EL2 concentration along the wafer showed the typical W-shaped distribution. It is proposed that the EL2 origin is due to a complex lattice defect growing in the presence of stress. This was supported by the finding that the 323 7 7 EL2 level is created in GaAs during mechanical deformation and that antisite defects form during plastic deformation by 7 8 dislocation climb . It was then proposed that the EL 2 is the 7 9 As(Ga) antisite or a complex containing the antisite In addition, the s t o i ch i ome t ry of the melt in LEC GaAs is shown to correlate with the SI properties of the material with a transition from SI for As rich melts to p-type material at the 8 0 — 8 2 critical As concentration of 0.475 atom fraction . This transition is also observed during annealing of SI undoped Ga 8 3 rich melts The proposed reaction leading to the antisite through dislocation climb is : V(Ga) + As(As) > As(Ga) + V(As) The most striking correlation of EL2 level distribution with 8 4 8 5 EPD distribution ' is shown in Figure 1.1 (a.b) where the EL2 level distribution is shown for wafers from the seed end (a) and from the tail end (b). The typical four-fold symmetry of dislocation correlates perfectly at the seed end. It is interesting to note the two following aspects : first four local minima are observed in the <110> and second, at the seed end a two-fold symmetry appears with a mirror plane coincident with a (110) plane. Two-fold symmetry of dislocations at the seed end 3 3 has been reported by Weinberg . Nevertheless, the fact that the EL2 level is not directly proportional to dislocations has led to the proposal that the EL2 concentration is proportional to the 8 6 dislocation density and to the amount of dislocation climb . At 90 91 94 95 present the antisite has been found ' ' ' and shown to be 324 Figure 1.1 EL2 contour plot taken at the seed of a(100) 3-in-diam LEC GaAs crystal. The solid dots indicate positions where absorption measurements were made, and the contour lines were drawn to be consistent with the measurements. The number assigned to each cogtoug indicated the EL2 concentrator) ^in units of 10 cm (i.e., 2.05 means 2.05 X 10 cm ). The cross-hatched areas indicate local maxima. 325 Figure 1.1(a) EL2 contour plot at the tail end of the crystal depicted in Figure 1(a). This EL 2 distribution has a mirror plane coincident with a (110) plane. 326 9 7 9 3 stable at high temperatures ' and believed to be the origin ft 9 of the EL2 level . The enhanced concentration of EL2 level at dislocations has also been attributed to the aggregation of EL2 centers similar to 8 7 a Cottrell atmosphere . Whatever the mechanism for [EL2] enhancement is, it has been demonstrated that dislocations play a fundamental role. This is based on the direct observation of the 8 8 EL2 concentration at single dislocations However, there is not complete understanding of the semi-insulating mechanism in GaAs. 9 6 9 7 1) There is more than one midgap trap (an EL2 "family") 2) A midgap level related to oxygen with larger electron capture cr os s -sect i on has been identified98'". 3) The characterization techniques of EL2 have been inconsistent100,101, specifically, different response of the mid gap level have been obtained in heat treated 10 2 horizontal Bridgeman grown and LEC grown GaAs 4) There is a close correspondence between the increase in sheet and resistance and an increase in EL2 concentration (opposed to what is expected) in dislocation free . 10 3 GaAs The EL2 level has been associated with undesired backgating effects108,110. This reinforces the need for a complete understanding of the origin and properties of this level. 327 The presence of carbon in GaAs has been shown to affect 104 directly the threshold voltage of implanted FET (carbon concentration in LEC GaAs depends on the water content of the 10 5 encapsulant ). Carbon is found segregated at the cell walls of 106 the cellular arrays of dislocations Also it has been shown that oxygen, chromium and silicon impurities in undoped GaAs segregated to the dislocation clusters43. I .2 Dislocations and Fermi Level Dislocations in GaAs have been correlated with a shift of 12 6 12 7 Fermi level by Lagowski ' . This is illustrated in Figure I.2(a-c). In Figure 1.2(a) the effect of the melt composition on dislocation density is shown for lightly doped GaAs. Note that the minimum occurs above 0.500 atomic fraction. The shift is within the uncertainty of the measurements. In Figure 1.2(b) and (c) the Fermi energy and dislocation density are shown as a function of carrier concentration. The concentration of dopant is below the impurity hardening regime. Note that n-type doping reduces the dislocation density. This effect is accounted for by a vacancy condensation mechanism in which the Ga vacancies may be the critical native defect. The As vacancy which can also be important in dislocation generation, can be created upon migration of the Ga vacancy to the neighbouring As site, V(Ga) + As(As) > V(As) + As(Ga) In this reaction the As antisite vacancies are midgap levels (V(Ga) As(Ga) is also created. Both in the lower half and V(As) in 328 I06 Arsenic fraction m melt,Xfl5 0.495 0.500 0.505 0.510 I02 0 p-typ€ • n-type _|_ 611 613 615 617 619 621 623 Arsenic source temperature,TAs(°C) Fig. 1.2 a Dislocation density vs arsenic source temperature (melt stoichiometry) for lightly doped n-type and p-type GaAs I017 I016 I016 I017 Carrier concentration (cm-3) I0« Fig. 1.2 b,c Dislocation density vs 300-K free-carrier concentration for crystals grown from optimum melt stoichiometry and under optimal thermal stresses. Upper poition shows the corresponding values of the Fermi energy at 1100-K. 329 the upper half of the band). Their occupancies will depend on the Fermi level. The amount of neutral states would determine the migration of the vacancies and the ability to coalesce into dislocation loops. The downward shift of the Fermi level caused by the n- to p-type transition increases the fraction of neutral V(Ga) states. In lightly doped material the above reaction takes place between charged (occupied) states V(Ga)++ + As(As) > As(Ga) + V(As)+ + 3e-The rate of reaction decreases with the third power of electron concentration. Since dislocation density increases upon transition to p-type it is assumed that the V(Ga) migrates rather than the V(As) since the reaction involving this type of defect would lead to the opposite behaviour with the transition from n-to p-type material. The exact mechanism for vacancy condensation, however, is not known. 330 APPENDIX II Integration by Parts of the Element Equation for the Temperature  Field A useful formula for the integration by parts can be written as [210] ,(e) N P , dD (e) D (e) N, £ P dD(e) + (e) NPn^dS(e) (II.1) and ' N 1 ( P), dD(e) (e) P N. P dD(e) + D (e) N P n dS *C * (II.2) (e) Where N and P are given functions of £ and p , and the comma indicates derivative, n^ and np are director cosines of the normal n to the elemental surface dSv 'in the £ and p directions respectively. The integral to integrate by parts is : (e) 1 3p2 36 326 2V P 3p 3C 33 dD (e) 0 ( I I . 3 ) Using the above formulae for integration by parts, we get D N . (e) 1 1 P 3 33 P 3p 3p .(e) dD (e) 3N, 33 dD(e> + D (e) 3p 3p _33 3p n dS (e) (II .4) 331 and 3 33 (e) 1 3? 3£ dD (e) 3N. 33 , . dD(6) (e) 3C 3C 33 N (e) 1 H n^dS (e) (II.5) Substituting in (II. 3), it is obtained that 3Ni 33 3NA 33 33 - 2V N (e) 3p 3p 3C 3? 1 3C dD<e) + 33 N . (e) 1 3p 33 3? n^ dS (e) Using the boundary condition (7.3) we finally get that 3N. 33 3N 33 . + + 2V N 33 ,(e) 3p 3p 35 3C 1 3? dD(e> + + r. (e) N.h(z) 3(6) dS(6) = r l 0 N.h z 3 z dS (e ) (e) 332 APPENDIX III Evaluation of the Matrix Elements for the Temperature Field a) Evaluation of K, ij Making the following substitutions in Equations (7.10) dD(e) = 2 TT pd pd £ 3L. i_ 3p 3L. c . 1 2 A 35 2 A we get 2 TT = ~ (b.b. + c.c.) Tij (2A)2 1 J 1 j p dpdC + 2Vb j 2 TT (2 A) J L;.pdpd 5 Substituting p = I p,L. and using tables for the integrals of k = l k k the natural coordinates we get : i J Trp(e> A2 *Vhl (e) (b.b. + c.c.) + 1 [ 3 p^ ' + p. ] ij ij 6 J where (e) ( I P, ) /3 i = l 333 b) Evaluation of K "ij These elements are evaluated for three different orientation of the external surface of the peripherial elements : i) vertical surfaces (parallel to the crystal axis) ; ii) tilted surfaces (inclined with respect to the axis) and iii) horizontal surfaces (perpendicular to the axis). i) Vertical surface Consider an element like the one in the figure The only contribution to the integral (7.10c) comes from the surface i-j . The elemental surface area in terms of the local z k r variable £ is dS (e) = 2TTpd£ with p = constant. Therefore K = rQh(z) L . L . i J dS (e) = rQh(z) 2iTp L . L . d £ i J H . . i J 0 where 1. . is the distance between nodes i and j. Between these i 3 two nodes the natural coordinates can be written as 334 Substituting and integrating we get i * j i j 2TTh(z) rQp i = J ij 1 . 2TT h(z) rQp LI ii) Tilted Surface For an element with tilted external surface the elemental surface area can be written as (e ) 1 dSie; = 2TTp.dc-where £ is the local coordinate as shown in the figure Then ij 2 TT rQh(z) ij L.L.nd; i J ^ 335 p can be written as a function of p^ , p^ and the local i coordinate £ as Pi + < Pj " P^ 1 . . i J Substituting p , and for the above expression and integrating we get [ KH ] 1 . . 27Th(z) r U~ 12 3pi + pj p . + p. i J p . + p. i Mj pi + 3pj iii) Horizontal Surface For an element with horizontal external surface the elemental surface area can be written as The natural coordinates are also written as L. = 1 l i J and 336 l J —c Approximating p by p = ( Pj + Pj ) /2 KH = 2 TT rQh(z) p i j (S) r l ij LJLJ dp Integrating, the elements are i = j K. ij 2TT rQh(z) p (S) i * j i j 2 TT r h(z) p (S) —^ 6 c) Evaluation of Elements K Making the same substitutions as in b, for dS^e^, L. and L. i 3 and p in (7.10d) for the three cases and interating we get i) Vertical Boundary Element TT h(z) rQ 6a(z) P lu ^ 1 ii) Tilted Boundary Element {KA> = TT h(z) rQ 6a(z) (PA + 2p) JLL 337 iii) Horizontal Boundary Element <V " *h<«> r0 6a(z) Plij 1 I 338 APPENDIX IV Evaluation of Stiffness Matrix Elements for Linear Elements Substituting the expressions for [B1] and [C'] and multiplying the four elements of the stiffness matrix of element e are : 1 b L b L k = [(1 -v) bb. +V2A L-A- + v 2 A —+ (2Ar 1 J p p L L . 1 - 2 V + (1 - V) (2A) LrJ— + c c ] pdpdC 1 c .L . 1 - 2 v k1? = r [vb b + v2A —1-J- + c.b ] pdpdc: 1£ (2A)Z 1 J p 2 1 J 1 C iLi 1-2V k01 = [vb .c. + v2A + c.b.] pdpd£ 21 (2A)2 J 1 p 2 J 1 , 1 1 - 2 V k22 " 77772 Id - V) c.Cj + — b.bj] pdpd£ I 2A } 2 The force vector components are 339 Also the initial force vector is 2A b. + 2 A 1 L . l • p dpdC Using tables for the integration of area coordinates we get 1 - 2V p k = [(1 -v) b b + c c ] + 11 1 J 2 1 J 4 A V + (b . + b . ) + (1 - v) A LiL.1 dPdC 1 - 2V P v k = [vb c + cibJ + c-j Xd 13 2 J4A 6J 1 - 2V V k21 = [VbiCi + Cibi] + Ci dl 31 2 J 1 4A 6 1 1 - 2V k22 = ni - v ) ciCj + b.b J 2 1 j 4 A 340 APPENDIX V Quadratic Element Calculations The stiffness matrix for the quadratic element is a 12 x 12 matrix with three different regions I [k1 ,13 [k1 ^ 22 [k']23 Symmetri c [k']31 [k- j 32 [k1 ^33 [k']41 [k1 ]42 [k' [k ,44 [k«]45 tk' [k']51 [k1 ^52 l~k' j 53 [k ]" [k']55 [k1 j 56 [k']61 [k- j 62 Ck' ^63 [k ,64 [k']65 [k' j 66 111 11 the [k •jqp are the 2 X 2 matrices relating nodes q and The displacement vector for any element is a 12 X 1 matrix as 1 (e ) well as the force matrix {F } o w. 11 12 (d> (e) w. 61 62 341 The Interpolation functions for the element are built from the area coordinates as 'i + 3 L. (2Li - 1) 4LiLi+l i = 1,2,3 and i + 1 has module three ; i.e. 1=3 1+1=1 (e) The form of the [B']v matrix for node q depends on the value of for q < 3 4 L 2A Lq <2Lq - *> (4 Lq - 1) and for q > 3 [ B 1 ] (e) 3+i 2 A Vi + l + Libi+1 2 ALiLi + l c . L . „ + L . c . , l l+l l i+1 c . L . , + L . c . . l l+l i l+l b . L . , + L . b . . l l+l i l+l again i+1 has module 3 342 Using these [B1] matrices performing the integrations we stiffness modal matrix elements in Equation (7.25c) operating and get the following values for the in the three regions. a) Region I, nodes 1 to 3 Case 1 q = p 11 [(1 - v) bqbp + l-2v Z p + 2p v c c ] 3- + (b + b ) 2 P q 10(2A) 15 L (2L - 1) L ( 2 L - 1) + (1 _v) |ji /_a a E p_ P1L1 + P2L2 + P3L3 dLldL2 ,qp 12 2v ( v b c + q p Z p + 2p V c . . n _q . p c b ) ( =*—) + K-Q P 10(2A) 15 .,qp '21 2v [ v c b + q P c b ] P q E p + 2p Kn 10(2A) vc. 15 22 2v = [(1 - v) cqcp + b b ] q P Z p + 2p Mn z_q_ 10(2A) 343 Case 2 q * p V >qp  kll 1-2V [ (1 - v ) b b + q p c c ](• p q 1 p + p + p n _g ( 30(2A) v 30 (b + b ) + q p L (2 L - 1) L (2L - 1 ) + (1 -V) |J| / 9 9 E E dL.dL. P1L1 + P2L2 + P3L3 12 12 1-2V Zp+p+p [vb c + c b ] ( E q~) q P 2 q P 30(2A) V c 30 . ,qp 21 2v [ vc b + q P I p + p + p c b ] ^ —9—) P Q 30(2A) v c 30 ..qp '22 2V [(1 v ) c c + q p I p + p + p b b ] (—-S ^ -3-) Q P 30(2A) q = j + 3 b) Region II, Nodes q, p > 3 p = i + 3 Case 1 i = j kl3l+1' 3 + J = — 1 li + 2v(bi + bi + l + bj + bJ + l> + 11 15 2 A 1 - 2V I' + + 16 (1 - v ) I. 2 2A 344 ,3+i, 3+j 12 2 15 V I 14 1 - 2V I 2A + 2V <Cj + Cj + 1> + 41 2A ,3+i, 3+j '21 V I 41 1 - 2v I 15 2A + 2V (ct + c1+1) 14 2A .,3 + i, 3 + j 22 15 (1 - V ) H + 1 - 2v 1 I 1 2A Where ; = 2b.b. (Zpn + 2p.+1) + 2bi+1bj+1 (Epn + 2Pi) + bibj+l (ZPn + Pi+1 + Pj> + bi+lbj (ZPn + Pi + Pj + 1> J| 1 1+1 ' J+1 dLldL2 ZpnLn l14 2biCj (Zpn + 2pi+l) + 2bi+lCj+l (ZPn + 2pi> + biCj + l (Zpn + pi + l + pj> + bi + lcj CPn + pi + pj + l) I' = 2c.b, (Zp + 2p, .) + 2c. ,b. , (Ep + 2p.) 41 I j * *n » Ki + 1 l+l j+1 v Kn wi + Cibj + 1 (Zpn + Pi + l + Pj> + ci + lbj (JPn + Pi + Pj + i> Case 2 i * j * i + 1 j + 1 = i , 3+i , 3+j 11 15 2 A Ij + v(bj + 2b i + 1 2b V.1 1 - 2v I ' 2A + 16 (1 - v ) I. ,3+i, 3+j 12 VI 14 2v 15 2 A + V(2Cj + c ) + 41 2A 3+ i, 3 + j Vi 21 15 41 2A 2v +v(c. + 2c. ,) + v l l + 1 ' 14 2A 3+1, 3+j 2v 22 15(2 A) (1 - V ) I£ + Where I" 1 bibj ( Zp. pi + l pj + l} bi+lbj+l (£ P. pj) + b.b. , ( Zp + p. . + p.) + 2b. ,b. ( Zp + 2p.) 1 j+1 v Kn Ki+1 Kj 1+1 j Kn Ki 346 lU = biCi (Zpn + Pi + 1 + Pj + 1> + bi + lCi + l (Zpn + Pi + Pj) + + Vj + l (ZPn + Pi+1 + Pj> + 2bi+lCi (ZPn + 2Pi) I" = c.b. (Zp + p. . + p. ,) + c. „b. . (Zp + p. + p.) + 41 i j Ki + 1 Kj + 1 i + l j+1 Kn wi Hj + Cibj+1 (Zpn + P1+l + Pj) + 2ci+lbj EPn + 2Pi> Case 3 j * i * j+1 i + 1 = j • 2 1 - V k,3+i, 3 + j = +v(2b. + b. + b. + 2b, ,) + 11 15 2A 1 1 1+1 J J+1 1 - 2v I4  + + 16 ( 1 _ v } j 2 2A "i n i tl3+i, 3+j 2 Vl14 . , . 1 ~ 2V *41 k 2 M + V(CJ + 2Cj + l) +  15 2A J J 2 2A II i II i u,3+i. 3+j 2 Vl41 , 1 2v *14 k2 = +v(2c + c )+ £l 15 2A 2 2A 2 1 - 2v k22 " (1 "V) I4" + J4' " 15(2A) 2 347 Where I"' = b b (Zp + P. + P ) + b. b. (Zp + p. + p.) + 1 i j v yn Mi + 1 + i + 1 j + 1 v yn Mi wj' + 2bibj + l (Zpn + 2pj> + bi + lbj (Zpn + pi + pj + l) I" 1 = b.c, (Zp + p. . + p. ., ) + b. c. , (Zp +p. + p.) + 14 ijKn *i +1 j + 1 l + lj + l'n "i Kj + 2b.c. . (Zp + 2 p. ) + b. ,c. (Zp + p. + p. ,) i j +1 K n Kj i + lj^n Ki "j +1 lll - Cibj (Zpn + pi+l + pj+l) + Ci+lbj (Zpn + pi + pj> + + 2c.b. , (Zp + 2 p. ) + c. . b. (Zp + p. + p. ,) ij + lVMn Kj ' l +1 j K n K I H j +1 ) Region III, Mixed Nodes q < 3 ; p = 3 + i ; i = 1,2,3 Case 1 q = i •a i+3 1 1 " v kl " \ *1 + V(3bq " bi + 2bi + l) + 30 2 A 4 1 - 2v c I4 + 4(1 - v) I5 2 A . n 1 V , 1 - 2 v c I ' *\V - — — V + V(2ci + 1 - c.) + -3—^-12 30 2 A Q 4 1 + 1  2 2A 348 •q.i+3 1 ' , 1 - 2V bq V ko; = c I, + 3Vc + u — 21 30 2A q 1 q 2 2A 'q,i+3 1 , 1 - 2v , K H = (1 _ v) c , + b i 22 30(2A) q 4 2 q 1 where I1 = (3b. . - b.)Zp + p (4b. + lib. -D. ,b. 1 l+l l wn wq y l i+1' Mi+1 I 1 A = (3c.,-c.)Zp +p (4c. + 11c. J - n . ,c. 4 i + 1 i Kn Kq i i + 1 Mi + 1 I L (2L - 1) L . L . . M » L»L DLIDL2 2 PnLn Case 2 q * i i + 1 * q •a 3+i 1 (1 " V) = b I. - v(b. - b. + b ) + 11 30 2A Q 1 1 1+1 Q l-2v c I" 9 + 4(1 - V ) IK 5 2 A •q,3+i 1 V ba I4" , , 1 - 2^ Cq V k = - V(c + c ) + l& 30 2A 2 2A 349 1 vcl" l-2v b I " K '3 + 1 = — —a_J_ _vc + g_j_ 21 -~ q o 30 2 A 2A .'q.3 + i 22 1 - 2V 30(2A) (1 - V) cq I4 + B 1 1 q i where (b. - b. ) (4p -Zp ) - (p.b. , + p. l i+l q Kn Ki i+l Hi+1 i (c, - c. (4 -Zp ) - (p.b. . + p. ,b.) v i i+l' v q *n wx i+l Mi + 1 I Case 3 i * q ; i+l=q .'q.3 + i 11 1 - v 30 2A b I + v (3b +2b. -bJ1)+ q 1 q l i+1 1 - 2v c 2A I 4 + 4(1 - V ) Ig ' q,3 + i 12 30 2A b r + v(2c. -c. ,)+ q 4 l i+l 1 - 2V c I "' 9 1 2A 350 q , 3 + i 22 30(2A) (1 - V) c i" ' + q 4 2V q i where •i i I 1 (3b, - b. .) Zp + p (lib. + 4b. ,) - p.b. , i l+l Kn q 1 l+l Ki l+l I4" - <3ci " ci+l)ZPn + Pq (11Ci + 4ci+l> - PlCi+l d) Calculation of force matrix elements F ' 0 q Substituting [B'] and [C1] into (7.26c) and performing the multiplication and integration we get : for q < 3 <Fo> l 3 P q q for q > 3 <Fo> i + 3 b,(Zp + p. ,) + b. , (Z p + .) + 2A i x n i + 1 l +1 "n l c.(Zp + p. .) + c. , (Zp + p.) in "i +1 i + l^n Ki 351 e) Strain Components The strain components as a function of the reduced displacement for the calculation of stresses at node q in element e are Case 1 q < 3 1 [3b u, - b , u . ,-b.u, p,q ~2~E~ q red,q q + 1 red,q+1 q + 2 red,q + 2 + 4b , u. + 4b . u„ ,1 q + 1 3 + q q-1 3+q-lJ u , P _ red , q E-Q " P q e _ = 1 [3c w , - c , w . « - c „ w „ + 2 A q re(J.q q+l red,q+1 q + 2 red,q + 2 + 4c ., w . n +4c . w . „ ., 1 q+1 red,3+q q-1 red,3+q-l ' _ = 1 [3(c u . + b w . ) - (c. , u , p^'q 2~A~ q red.q q red.q l+l red,q+1 + b w ) — (c u +b w ) + i + 1 red,q + 1 q + 2 red,q + 2 q + 2 red,q + 2 + 4(c ., u , „ + b Hw , „ ) + q+1 red,q+3 q+1 red,q+3 + 4(Cq-l Ured,q-l+3 + bq-l Wq-l+3)] 352 Case 2 q > 3 ; q = 3+1 ; i = 1,2,3. £P3 + 1 " 2V[bi "red,! + bi + l "red,i + 1 ~ bi + 2 Ured,i + 2 + + 2(b. + b. ,) u JJO+ i i+1 red,i+3 + 2bi-l (ured,1+1+3 + Ured,i-l+3)] r o.i = 1 [c.w j i + c. , w ,..-c.„w , , « + s.,3 + i 2 - 1 i red.i l + l red,i + 1 i + 2 red,i + 2 + 2 (c . + c . H ) w _ , + l l+l red,i+3 + 2c (u +w i l i-1 v red,3+(i + l) red,3+(i-1)'J Y ^ • o = 1 [c.u ,.+b.w , . + c . .. u . . « + p4,i + 3 2 ^ l red.i i red,I i + 1 red,i + 1 + bi + l Wred,i + 1 ~ (ci + 2 "red,i + 2 + bi + 2 Wred,i + 2) + + 2[(c. + c. ) u , . „ + (b. + b. H) w J . „ + i l+l red,i+3 l l+l red,i+3 + Ci-l(Ured,(i+l)+3 + Ured,i-l+3) + + bi-l (Wred,3+i+l + Wred,3+i-l)] ] 353 APPENDIX VI Calculation of Resolved Shear Stresses The 12 slip plane and direction combinations for the {111} <011> glide system are given in Table V.I. For a crystal grown in the [001] direction vhich is coincident with the Z-axis, the three directions in the (111) plane, for instance, are shown in the Figure. Similar figures for the other three (111) planes can be drawn The first step, for operational reasons, is to project the stress tensor onto a cartesian coordinates system. The second step consists of rotating the cartesian system in order to make one of the axis coincident with the glide direction and any of the other axis perpendicular to the glide plane. The shear stress on that plane and direction is the corresponding resolved shear stress to be calculated. For the calculations the definition of the tensor is used. A set of quantities T, with respect to a x 354 coordinate system is a tensor if under a rotation of the system the quantities T, transforms as T = M.T.M"1 where M is the operator performing the rotation of axis which is defined positive when is done in the clockwise direction. a) Stress in Cartesian Coordinates The transformation from a cylindrical system of coordinates (P, 6, z) to a cartesian (x, y, z) is identical to a rotation-9 over the z-axis as shown in Figure. Z e The matrix for this transformation is cos6 -sine 0 Mz (-6) = sine cos e 0 0 0 1 for this type of transformations the inverse operation is 355 and therefore oxx oxy axz oyx oyy ayz axz ayz az z which gives -0) = Mz<~e> = Mz(e) cos6 -sine 0 = sine cose 0 X 0 0 1 -cos6 s i n9 0 X -s in6 cos8 0 0 0 1 a XX = °P cose + ae sin26 °yy ' = °P sine + a6 2 cos e a z z 0 z °xy ' sin e cos e (OP " °e a xz = TPz cos6 ayz ! = TpZ s in6 p 0 pz 0 b) Sample Calculation of the RSS in the (111) Plane and  [1101 Direction Two rotations are performed : with the first rotation over the z-axis by an angle a = 3 / 4 IT the rotated system (x ' , y', z' = z) has the x'-axis coincident with the [110] direction ; with the second rotation over the x'-axis by an angle 3 such that cos 3 = /3/3, the new system (x" = x', y" , z") has the z"-axis 356 perpendicular to the (111) plane. In this case the desired stress component is o . The matrix for the combined rotation in the order the rotation is performed, is calculated as M(o . 3 ) Mx, (3 ) Mz(a ) with and Mx,(3) 10 0 0 cos B sin3 0 -sin(3 cos S Mz(a) cos a -s in a 0 s i n a 0 cos a 0 0 1 Multiplying both matrices to obtain M(a, 3 ) and performing the rotation of the tensor the desired a „ component is : X z x" z (o -a ) sin a cos a sin 3 + xx yy 2 2 + a (sin a sin 3 - cos a sin3 ) + xy + a cos a cos 3 + o sin a cos 3 xz y z Similar calculations are The performed rotations for each case are listed repeated for the and corresponding in Table VI.1 . other 11 projections, calculated component 357 Table VI.1 Rotation performed for the calculation of the RSS ROTATIONS | Slip | Slip | 1 2 (Calculated plane direction Component |axis | angle |axis | angle | | 110 z 3/4 TT x 1 g x"z" 111 Oil x TT/4 Z* 3 z"x" 101 y TT/4 X' 3 x"y" 101 y 3/4 TT x' -3 x"y" Til Oil x TT/4 Z' TT/2- 3 y"z" 110 z TT/4 X' - 3 x"z" Oil x -TT/4 z' -(TT-3) x"z" 111 101 y -3/4 TT Z* -(TT-3) X"Z" 110 z -TT/4 x' 3 x"z" Oil x -TT/4 z ' - 3 x"z" 111 101 y TT/4 X' TT/2-3 x"z" 110 z -3/4 TT x ' - 3 x"z" When the values of the cartesian components of stress are replaced in terms of the cylindrical components the twelve RSS are : 358 Table VI.2 Resolved Shear Stress component in the <110> (111) slip | Plane |Direction| Resolved Shear Stress [110] /6 <-(0 -0O)(cos2e-sin26 )+x (cos9-sin0)} —- p 0 pz' b (111) [Oil] . o 2 /6 {-(0 sin 0+o„cos 6)+ 6 p 9 + ° _(0„-Ofl)sin0cos9+T cos9 } z z p 0 pz [101] .— 2 2 /6 {o co 0+a„sin 9-a +(o -aQ)sin6cos0-a sin0) —p 0 zz ' p 0' pz b [101] 2 2 /? {o cos 8 + onsin 0-a -(a -a„)cos0sin0-T sin0} —q P 0 zz p 0 pz [Til] [Oil] 2 2 /6 {-(o sin 0+aQcos 0)+ 6 p 0 +0 +(o -o„ )cos0sin0-T cos0} zz p 0 p z [110] 2 2 /6~ { - (o -on ) cos 0-sin 0 )+T (cos0-sin0)} g P o p z [Oil] /6~{-(a sin20+aQcos20 ) + ~6 P 6 + o -(0-OQ )cos0sin0-T cos0} zz P o P z (111) [101] 2 2 /6~ {o cos 0+a„sin 0-~6" P 6 -a +(o.-ofl )cos6sin0+T. sin0 } zz P 0 P z [110] 2 2 /?T { - (Op -Og ) ( cos 0-sin 0 )+xpz (cos0-sin0 ) } 6 Cont . / 359 Cont./ [Oil] /~ (-(o. sln26+QQ cos 0 ) + +0 +(a -o„ ) cos0 sln0+T cos8 } zz p 0 p z (111) [101] /6~ (o cos 0+aQsin 0-~6 P 6 -a -(a -on ) cos0 sin©+T sin0 } z p 0 p z [110] /6" {-(o -aQ ) (cos20-sln20 )-T (cos0+sin0)} 6 P y P z 360 Using trigonometric relations they can be put in a more elegant form as follows : Table VI.3 Compact form of the RSS from Table VI.2 |P1ane|Direction| Resolved Shear Stress | Mode | [no] /6~{-(0 -Oa ) cos26 +/2T rcos (8 +TT/4 ) } ~6 P 6 PC I a (111) [oTi] /6~{-(a -Or, )/2sin6 sin (8 +TT/4 ) + (ar -0Q ) + ~6~ p 6 ? 6 +a _cos8 } PC 11 a [ioT] /6~ (a -Or, )/2cos8 sin (8 +TT/4 ) - (ar -oQ ) -~"g~ P D Co -T rsin8 } PC 111 a [ToT] ~ { (o -oQ )/2cos6 c'os (0+TT/4 )-(oc-aQ ) -— p 0 C 8 -xpCsin0 } IV a [in] [oTi] /6"{(o -0Q )/2sinecos(0+Tr/4) + (a -aQ ) -6 P o Co -T COS0 } PC V b [no] /~" { - (Op -Og ) cos20 +/2Tp^ COS (0 +TT/4 ) } 6 I a [Oil] /6~ {-(Op-Og )/2sin0sin(0+Tr/4) + (a^-Og ) -6 -Tp^ COS0) 11 b (TTi) [ToT] /i~ { (OD-OQ )/2cos0sin(0+Tr/4)-(Or-Og ) + 6 +TpCsin0} 111 a [iTo] /6~{-(0 -O ) COS20+/2T COS(0+TT/4)} ~e p 6 PC I a Cont./ 361 Cont . / [Oil] /6~{(o -0. )/2sin6cos(e+Tr/4) + (o -a.) + 6 p y Co +T _cos6} PC V a (111) [ToT] /6~{(0 -0Q)/2cos6cos(e + TT/4)-(ar-0Q) + 6 P o Co +T rsin8) PC IV b [!!o] Zi"{-(0 -0 )cos29-/2T COS(6 + TT/4)} 6 P 9 PC I b 362 APPENDIX VII Temperature Field During Cooling The problem to solve is given by the following equations. 1 3 33 323 33 ( p ) + — = r (VII.l) p 3p 3p 3^ 9t I.C.1s at t* = 0 30 = P0 + Pl^ + P2 ^ (VII.2) B.C.1s at t* > 0 33 / = -h'3 (VII.3) 3p p = l 33 / = -h'3 (VII.4) 3? C=Ct 33 / = h'3 (VII.5) A solution is obtained by separation of variables i.e. 3(p. £. t*) = R(p) Z(C) T*(t*) (VII.6) -Tt* Morever, we assume T*(t*) a e , ¥ is constant. Substituting 3 in (VII.l) and operating we get 363 1 1 d dR(p) 1 d2Z(C) (p ) + - = -f (VII.7) R(P) P dp dp Z(C) dC 2 2 which is satisfied if each term is constant, say -X and -y respectively. This gives the following ordinary differential equations and conditions d2R 1 dR + + R = 0 (VII.8) d(Xp)2 (Xp) d(Xp) with R such that dR / = -h'R (VII.9) dp p=l and d2z 2 — + y Z = 0 (VI I.10) dC2 with Z subject to dZ / = -h'Z (VII . 11 ) d£ ? = tit dZ /=h'Z (VI I.12) d? ? = 0 and also X, y and T must satisfy 364 (VII .13) The solutions of (VII.8) and (VII.9) are the Bessel functions of zero order, i.e. R(p) JQ(Xp) for all the A values satisfying the algebraic equation (see Appendix VIII) As shown in Appendix VIII, these functions constitute a complete and orthogonal set of vectors. The solutions of (VII.10) are the harmonic functions, in order to satisfy the B.C.'s (VII.11) and (VII.12) we choose as solutions Z (C) = A cosy K + B siny £ (VII.15) substituting in (VII.12) and (VII.13) we obtain respectively that X Jj (a) = h' JQ (X) (VI I .14 ) A Y (VII.16) B h and 2Y h tan Y CT = Y' (VII.17) 365 (VII.17) gives the condition to obtain the y-values. Moreover, the functions (VII.15) subject to (VII.11) and (VII.12) are orthogonal (see Carslaw and Jaegger). The general solution for 3 is therefore where the summation extends to all the A and y -values which are roots of (VII.14) and (VII.17) respectively. To obtain the coefficients C*( \,y ) the initial condition (VII.2) is applied and the inner product in the space-function is performed, that is 3( p- 1*) = Z C*( a.y) e 2 2 -(A2 + Y ) t* JQ(Xp ) [YcosyC+ hr sinyC] (VI I.18) fC t = C*(A,Y> dp (VII . 19 ) J 0 It is observed that C*(A,Y) can be written as C*(X,Y) = C1(X) C2(Y) (VII.20) After performing the integration the coefficients are found to be 366 CAX) = (VII.21) (h,<J + A ) JQ(A) and C2(y) =11/1 (VII.22) with 't (y 2 + h'2) + h' (VII.23) and II = siny^ {pQ + Pl(tit + h'/Y2) + P2(C2 - 2/Y2 + 2 th'/j2)} + C0SY4t 2 2 + — <-P0h' + Pj(l - h'Ct) + P2(2?t - h'Cj + 2h'/Y )} + Y Pnh' p, 2p h1 + 1 ^— (VII.24) Y Y Y Also the solution for 3 can be written as a product of two independent series _A2t* 3( p. £ . t*) = 2h1 [ Z C ( A) e Jn(Ap)] A 1 0 .Y2t* {Z C2(Y) E [YC°SY £ + h'siny?]} (VII.25) Y 367 APPENDIX VIII Analytical Solution of the Temperature Field The partial differential equation to be solved is : 323 l 96 323 33 3p2 p 3p 3£2 3C subject to the following Boundary Conditions 33 / = -hrQ3 (a) 2V = 0 (VII I . 1 ) 3p p= 1 at £ = 0 (b) (VII I . 2 ) at £ = Ct (c) Equation (VII.1) can be solved by separating variables 3 ( p,C ) = R(p ) Z(£ ) which leads to the following two ordinary differential equation d2R 1 dR + + R = 0 (VIII.3) d(Ap)2 Xp d(Ap) 368 d2R dZ — - 2V - X Z = 0 (VIII.4) d«T d£ Solutions of (VIII.3) are the Bessel functions of zero order R(P)•JQ (XP) Equation (VIII.4) is linear and homogeneous with constant coefficients with the general solution £ /V2+ 2 — EVV2+A2 V£ Z(C) = (Ax e> /V + Bx e ^V A ) eV^ where A^ and B^ are constant to be determined. The general solution is therefore 6 (p,C) = eVC ZJQ(Xp) [AA ek*C + Bx e k^5 (VIII.5) X 2 2 where = V + A . and the summation is carried out over all the possible values of From (VIII.2a) we get that XjJf(X) = hrQ JQ(X) (VIII.6) from which the values of X are obtained. From (VIII.2b) and (c) it is obtained that 369 and 1 = I J0(Ap) [Ax + BA ] A VCt kACt -\lt e I Jg(Ap) [A^ e + e ] A (VI11 .7) (VIII.8) In order to calculate A-^ and from the above equations it is necessary to show that the functions J(Ap ) from (VIII.6) form a complete sub-set of orthogonal functions. To do this let us calculate the coefficients for the expansion of the unit function in terms of J(kp), that is 1 p J0(kp) d = z 0 A J0<Ap) P J0(kP) dP(AA + V where 1 was substituted by (VIII.7) and p is used as a weighing function. Using properties of the Bessel Functions the integral inside the summation in (VIII.9) can be written as for A * k Jn(Ap)J (kp) pdp = 0 0 0 A2-k2 [kpJQ(Ap ) JQ(kp) ApjQ(k p) JQ(Ap) ]Q — [-kJQ(A)J1(k) + AJQ(k)J1(A)] A"-k 370 1 [-JQ(X) hrQJ0(k) + JQ(k) hrQJ0(X)] = 0 which shows the JQ(Ap) are orthogonal. The last two equalities were obtained using the relation between derivatives JQ(Ap) = -J1(Ap) and Equation (VIII.6). For A = k and following the same procedure gives the norm as f1 u 2 2 h r JQ2(Ap)pdp = \ jl + 1— j JQ2(A) (VIII.10) .0 X Knowing the Jn(Ap)'s are orthogonal with a norm given by (VIII.10) from (VIII.9), we get after integration that 2hr AA + BA 0 (A2 + h2rQ2) JQ(A) VII . 11 Another equation for A^ and is obtained from (VIII.8) This is valid for any p , which means each term should cancel, i.e. M t _ kA ?t A. e +B, e u = o (VIII.12) . * Mathematical Methods in Physics. J. Mathews and R.L. Walker. W.A. Benjamin Eds. N.Y. 371 From (VII.11) and (VII.12) we get 2k t% Ax = -Bx e X t (VIII.13) and 2hr0 1 BA = —2 h2 2, . . "j 2kAgt, (VIII.14) (A + h rQ ) JQ( A) (1 - e A<=t ) Finally the complete expression for the solution for the temperature field can be written as vc r VAP> e(p,C) = 2hr e " Z A (A + ITr^) JQ(A) kA^ - 2kA^t kA? - e e (VIII.15) , - 2kACt 1 - e 2 2 2 with k = V + A and the A's satisfy A A Jj(A) = hrQ JQ(A) (VI11.16) 372 APPENDIX IX Plane Strain Analytical Stresses The plane strain analytical solutions for stresses are derived assuming there is only radial displacement and allowing an axial displacement in order to satisfy the condition of no axial traction at the ends of the cylinder. In non-dimensional variables these solutions are 3 Pd p 3pd p (IX.1) , 1 3p d p + rP 3pdp - 3 (IX.2) A = 2 3 p d p - 3 (IX.3) with a conversion factor given by a E(TMP " V dim 1 - V (IX.4) A. Axisymmetric Thermal Field For the general temperature field given by Equation (VIII.15) the stress components, after integrating by parts, are 373 ve hro op = 2hrQ eV^ Z Kx( £) [ f X X Jj(Ap ) Ap J0(A) (IX.5) o„ = 2hr ev^Z K U) { f- + [ * Jn(Ap)]> (IX.6) 9 ° A A Jn(X) Xp ° VC r 2hrQ ^ X X V Ap) J0(A) (IX.7) where KXU> _2k>^i- 2 22 (1 - e A fc ) (A + iTr*) (IX.8) To obtain these expressions the following integral was used ,1 JQ(Z) ZdZ = J (Z) Z (IX.9) The evaluation of stresses at the center to the cylinder is accomplished by using the following limiting values JQ(0) J, (Ap) 1 and 1 / Ap p->- 0 1/2 (IX.10) 374 B. Radial Temperature Field For a temperature field given by 3 stress are immediately obtained as : p the values of op 1 / 2 1(P 1) (IX. 11) ae = 1 (3 P2 - 1) (IX.12) 1/2 (IX.13) 375 APPENDIX X ANALYTICAL AS ISYMMETRIC SOLUTIONS FOR STRESSES X . 1 Axisymmetric Temperature Field A) Love's Potential * 3 The reduced Love's potential L = L/EaS^r satisfies the i o biharmonic differential equation : 2 2 * V V L =0 i a where a od p a p a v with L , by symmetry, only a function of p and £ . Performing a cosine finite Fourier transform on the biharmonic equation we get A 2 2 ,2 , , 2 2 Id n TT d Id n TT dp pdp £~ dp" pdp £t 2~) Lc (n,p) where by definition the following relations were used Ffc {L (£,P)> S Lc (n,p) £t „ n TT£ L (£,p)cos d£ and Ffc <• a2L*(£, P) a7 2 * Ct 9^L n TT £ cos a£' d£ 376 (-1)' 3L 3£ 3L Ct 3C 2 2 n TT T L c(n-P) with the assumption that 3L 3L 3C 3C 0 The antitransform is performed by summation of the following series 1 „ 2 °° „ nrrC L (Cp) = L (0) + Z L (n.p) cos A fairly general solution of the biharmonic equation which is the modified bi-Bessel equation, is obtained as a combination of the modified Bessel functions of order zero and one as L (n,p) = A(l ) I (1 ) + B(l ) (1 p) 1,(1 p) c ' n on nnK 1 nK with ln = nTT/^ what can be verified using the following recurrent relations among derivatives of the modified Bessel functions x I'n(x) = x I^U) - x In(x) ; x I'n(x) = x In+1(x) + x In(x) ; 377 The functions are not included because they diverge for p -»- 0. The general form of the antitransformed Love's potential is therefore: L*U. P) = I [AUJ I (1 P) + B(l ) (1 p) 1.(1 p) c . non nnin £ t n= 1 cos 1 £ n with Aj and constants to be determined from the boundary n n conditions. B) Goodier's Potential * 2 In terms of a reduced potential <b = <J>( l-v) / (1 +v) aS^r and the nondimensional quantities p , £ , and 3 , the Goodier's potential can be accomplished by solving the differential equation 2 * 2 V <b = 3 ( p , £ ) where V is the harmonic operator in cylindrical coordinates and 3 (p . £ ) is the temperature distribution in the body given by Equation (VIII.15). The form of this function, a series of a products of functions J (A,p) Z (£) " A * of one variable, suggests that we can propose a solution for tb * as a series of functions (b such that A V 2 <bA* = JQ( A p ) ZA(£) Z (£) being the function Zx(£) = e [e - e e ] 378 Performing a sine finite Fourier transform on the differential equation such that 2 * 3 V fs 9?2 A, n , n 2 * 1 r^— sin 1 £ d £ 3£2 n 0 9<J)> in n-nn+1 *A ut) + 3? where A.n <|>x(p.C) sin ln £ d £ with the antitransform given by 4>A (£.p) £t n=1 We get the transformed differential equation as 3 1 3 <—•-—- l/) (n.p) = J0(Ap) Z (n) 3p p 3p Where it was assumed that 3<1> 34> 3£ 3£ for a particular solution. 379 Also Z, n (n) = Fo {Z, (£)} A, n s A The differential equation for (b. can be modified to get A» n d2 Id ln2 * J0(Xp) + —x) (b, „ (n.p) = — Z. n (n) d(Ap)2 p d(Ap) A2 X,n ' A2 A,n A particular solution can be obtained as O'x.n - A VAp) Substituting and using the following relations among derivatives of J^and : J'0(Ap) = - Jj(Xp) and JjUp) J"0(Ap) = - Jn(Ap) + Ap we get that ZA,n<n> \2 i 2,,2 A +1 /A n and zA,n("> ^A.n = " x2 + 1 2 J°UP) 380 Applying the antitransform , 2 oo z (n) *A<e.p> = - J0Qp> — Z 2'n 2 sin 1C £t n=l (A + ln ) The complete solution for <J> is therefore 4) (p.C) 2hr. Z A Jo( > A2(A2 + h2r2) JQ(A) 2 oo z (n) sin 1 £ E A .n ni Ct n=l (1 + ln2 /A2) (1 - e~2kACt) which can be put in a more suitable way as * 4hr oo 4) (p.£) = Z sin 1 £* € t n = 0 I J0(Ap ) JQ(A)(A2 + h2rQ2)(A2 + ln2)(l "2kA^t, e ) Z , (n) A, n C. Calculation of Constants Aj and and Stresses n n The non-dimensional components of stress derived from Love's and Goodier's potentials are 9 * 2 * 9 » 3 L 13 4) on(e.p) = [vv L —] + —(—-— - B ) M 3£ dp l-v 3p 381 9 t 1 9L 1 1 3<b oe(Cp) = tvV L ] + ( 3 ) 3£ P 9p 1-v p 3p 2 * 3 3 2L [(2 - v) V L + — 2 * 1 3 <J) ] + (• 3£ 2 J 2 3£ 1-v 3£ 3 ) PC 2 * 3 o * 9 L [(1 - v) V L - ] + 2 * i 3d> 3p 3£ 1-v 3p3£ Subject to the following boundary conditions o = x _ = 0 P PC and o p.= 1 E- 15" 5, £= o After derivating, the stress components are a X{<I0(1nP> tAln (2V - 1)] + n 1 + tBl (1np> - " n lnp 2 2 2hrQ ZA,n(n) tJQ(Xp) 1„ A + -^(Apl/Apl 1_V ~2kl^t 2 2 2 2 (1 - e A *) (A+hr^ ) [1+(WA) ] JQ(X) sin 1 £ n* 382 2 V- f Il(1nP) 3 £sln 1C K JL_ A - io(lnp)(2V-l) Bj ] ln C n v 1 p n n t n 2hrQ Z (n) tJ^Ap) /Ap - JQ(Ap) (1 + ln2/A2)] + ^ ' 2k A (A2+ h2r02) JQ(A) (1 - e A *) (1 + 1^/A2) 7 sin 15 {(I0(lnP) [Aj - 2(2 - V, B ] n v n n + (1np) Bl > V + n VAP> ZA,n(n) 2hr + Z  1'V \ 2 2 2 ~2RACt 2 2 A (A% h rQ ) (1 - e A l) (1 + 1„ A ) } PC «t n I cos V {<I0(1nP) Blp + ll{1np) n v. n [Aj + 2(1 -y) Bj ]} ln + n n 2hr _ + " 1 I n ^ VAp> ZA,n(n) } A A(A^h2r02) J0(A)d + ln2/A2)d - e 383 These components of non-dimensional stress should be multiplied by E a 3 to obtain the stress in absolute units ; i.e. |o|J4 = F S „ | O I . For the calculation of stresses at the axis, dim a f p = 0, the following limits are used : 1/2 J^O) = 0 J0(0) I0(x) Jj(x) x 0 x 0 1/2 The evaluation of the constants A^ and n n is carried out by solving the system of linear algebraic equations resulting from the application of the boundary conditions a and T P = l PC = 0 p = i Ai n 1,(1 ) 1 n •] + B, [Il(ln) ln - (2V - 1) I0(ln)] 2hr0 1 2 Z K. (n) (l/ + hr ) 1-v 1 3 A,n " ° n Al W + Bl [I0(1„» Jn + 2(1"v) !l(1n)] n n 1 -v 1„ X 2 Z KX.n(n) 384 where Z>.n<"> -2k £ A2(A2 + h2rQ2) (1 + ln2/X2) [1 - e A t] Solving for Aj and Bj we get n 2hr = ~ — (Z K (n) (In(l ) [1 + 2hr (l-v)] + 2n A(l-v) ln2 X,n 0 n n 0 1(1 + 1 ^ Hn (hrQ + 2(l-v)) + 2hr (1-v)] } and 1 2h2r02 ZKX n(n) Jn Bl - 7 < - " " "TT- [ VV + > n A 1-v 1 hr„ n 0 where 2 li (1n> 2 * = 'o'V ln ~ 2 t2^ -v) • ln23 D. Calculation of Z X, n ZA.n<"> " Fs <V5,) f^t v£ -kAC _2kA?t kA? nTT e [e - e e ] sin £ d £ 385 n TT A t , ,,11 t ! - (-1) e A t , n t>, - (-1) e 2 2 n TT 2 2 n TT (V-kx)» • (v+kA)" + 2 . Analytical Axisymmetric Solution for (3 = p The same procedure form part 1 is followed applied to a 2 temperature field that depends on radius only, i.e. 3 = -p . A. Love's Potential The differential equation for the Love's potential is the same as shown in Part A, as well as the solution i.e. 2 °° L*(£,p) = Z [A I0(1nP) + Bl (1nP) ^(^P*!008 1 n^ 4 n=l n n B. Goodier's Potential The differential equation for the Goodier's Potential is 2 * * 2 * 3d) 1 dd> 3<J) 2 + 3p2 p 3p 3C2 After a sine finite Fourier transform is performed, we get 2 * * 9 frn 1 8<{>n .2 * 2 . — + - 1 4> -p f „ 2 . n * n H i 3p p 3p 386 where n TT sin £ d £ [1 - (-D ] The particular solution proposed is <b ap + bp + c , which n gives, after solving for a, b and c 4f 2 ' 0 , Finally, performing the antitransform 2 1 - (1)" 4 cb (p.£) = I ~ (p + 5-) sin 1 £ £^ 1 1 i n C) Calculation of Stresses and Constants Aj and Bj n n The stress components derived from both Love's and Goodier's potentials are : I {U0dnp) [A - B (2V - D] n n n Ajn 3 tl-(-Dn 2 + Xl(1np) [Bl(1np) " 7^]} V + < — + p > n n )> sin 1 ( J 2 Xl(1np) 3 — I {[—2-_A - ! (1 p) B (2v- 1)] ln3 + 5* X„P n n 387 [1 -<-l) ] (l-v) 1 1 n n P )> sln lnC °e Z({I0(lnp) [Aj - 2(2 -v) Bj ] + Ijd^) Bj } ln; Inn n 4 [1 -(-1)"] (1-v) „ ' 1 ~ 3 J Sln 1ni and 2 Z({I0{1nP)(1nP) Bl + Il(1np)[Al + 2(1_v) Bl ]} ln + (. n n n 2 [1 -(-I)"] p> cos 1 t% 1-v 1 2 ' n These non-dimensional components of stresses should be multiplied by Ea 3j. to obtain the stress level in absolute units ; i.e. <a>dim - Ea *f <°> The evaluation of the constants Aj and Bj is done as before, n n using the boundary conditions : i) o = 0 at p = 1 P Al [I0(1n> " Il!1"> ] + Bl [Il(1n> ln " <2v " l) n 1 n n 388 [1 - (-D ] (1 ) 1, + 1) for n = 2n (1 - v ) 1, *2 + 1) with 1 = n (2n + 1)TT ii) from T = 0 at P = 1 Al VC' + Bl [I0(1n*» C + 2(1 " V) *iO n n That is the only non-zero and come from odd values of the n n integer n Solving for A^ and Bj and droping the asterisk Al " - nM ^4 <W + V + 2 W D(l-v) ln ln and n D ( 1 - v) 1 1 1 n n n with 389 (2n + 1)TT 1 = and D = I02(ln) ln - [2(1 - v) + 1„2] n The evaluation of stress components at the axis of the cylinder is done using the limit values given in part 1 of In/P and I /p 390 APPENDIX XI COMPUTER PROGRAMS XI.1 Mesh Generator c C THIS PROGRAM GENERATES MESHES FOR FLEMT AND PFLEMS C C FOR CRYSTAL , CONE AND SEED OF BOULE C C DIMENSION RC(4000),ZC(4000),NODE(10000,3) DIMENSION NV1(100),NV2(100),HV(100),TINFV(100) DIMENSION NT1(100),NT2(100).HT(1OO),TINFT(100) DIMENSION NH1(20),NH2(20),HH(20),TINFH(20) DIMENSION LASTN(100),M(100) INTEGER NN,NBC,NBCL,LAST,LASTEL,NEL0,NELP.NODE INTEGER CBN,LL,M,LASTN.BNS,BNL,NNHCV INTEGER NV1,NV2,NT 1.NT2.NH1 ,NH2 REAL XL,HBC.WBC,RC,ZC,HEIGHT,BH,RAD,BLE,BHS,BWL REAL HV,HT,HH,TINFV,TINFH,TINFT C C C READ DIMENSIONS OF CRYSTAL AND CONE ANGLE C RE AD(5, 1010) SIZE,XL READ(5,1015) SEEDW,TAN,SH C C CALCULATE NUMBER OF BLOCKS FOR CYLINDER C NBC=XL/SIZE HBC=XL/NBC C C CALCULATE BLOCK WIDTH C NBCL=1 ./SIZE WBC=1./NBCL C C INITIALIZE VARIABLES C NNINT =0 NEINT=0 ZINT=0.0 LASTEL=NEINT LAST=NNINT C C GENERATE FIRST ROW OF NODES C DO 50 1=1,NBCL NOD=LAST + I RC(NOD)=WBC*(1-1) ZC(N0D)=O.O 50 CONTINUE C LAST=LAST+NBCL+1 RC(LAST)=1.0 ZC(LAST)=0.0 C C GENERATE REMAINING ROWS IN THE CRYSTAL C DO 200 1 = 1 ,NBC C DO 100 J=1,NBCL 391 NOD=LAST+J RC(NOD)=WBC*(J-1) ZC(NOD)=HBC*I 100 CONTINUE C LAST=LAST+NBCL+1 RC(LAST)=1.0 ZC(LAST )=HBC*I C C GENERATE CRYSTAL TOPOLOGY C DO 150 K= 1,NBCL NEL0=2*K- 1 + LASTEL NODE(NELO,1)=LAST-2*NBCL+K-2 NODE(NEL0.2)=LAST-NBCL+K NODE(NELO,3)=NODE(NELO.2)-1 C NELP=2*K+LASTEL NODE(NELP,1)= NODE(NELO,1) NODE(NELP,2)=N0DE(NELP,1)+1 NODE ( NE LP , 3)=NODE(NE LO,2 ) C 150 CONTINUE LASTEL=LASTEL+2*NBCL C C CYCLE FOR NEXT BLOCK C 200 CONTINUE C C C GENERATE NODES AND TOPOLOGY FOR CONE C M(1)=NBCL LASTN(1)=LAST C C CALCULATE NUMBER AND DIMENSION OF BLOCKS C HE IGHT=(1.-SEEDW ) *TAN CBN=HEIGHT/SIZE+1 BH=HEIGHT/CBN C DO 600 1 = 1 ,CBN RAD =1.-BH*I/TAN L = I + 1 M(L)=RAD/SIZE+1 BLE=RAD/M(L) LASTN(L)=LASTN(I)+M(L)+1 C NLINE = M(L) DO 500 J=1,NLINE C NOD=LASTN(I)+J RC(N0D)=BLE*(d-1 ) ZC(NOD)=BH*I+XL 500 CONTINUE C RC(LASTN(L))=RAD ZC(LASTN(L))=BH*I+XL C DO 540 J=1,NLINE C NEL0 = 2*<J -1 + LASTEL NODE(NELO, 1 ) = LASTN(I)-M(I ) + J- 1 NODE(NE LO,2) = LASTN(I) + J+1 N0DE(NEL0,3)=N0DE(NEL0,2)-1 NELP=2*d+LASTEL NODE(NELP,1)=NODE(NELO, 1 ) NODE(NELP,2)=NODE(NELP, 1 ) + 1 N0DE(NELP,3)=N0DE(NEL0,2) C 540 CONTINUE 392 IF(M(L).LT.M(I))GO TO 549 C LASTEL=LASTEL+2*M(L) GO TO 600 C 549 LASTEL=LASTEL+2*M(L)+1 NODE(LASTEL, 1 ) = LASTN(I )- 1 NODE(LASTEL,2)=LASTN(I) NODE(LASTEL,3)=LASTN(L) C C C CYCLE FOR NEXT BLOCK IN THE CONE C 600 CONTINUE C C GENERATE NODES AND TOPOLOGY IN THE SEED C BNS=SH/SIZE+1 BHS=SH/BNS BNL=SEEDW/SIZE+1 BWL=SEEDW/BNL LL=CBN+1 LAST=LASTN(LL) PHT=XL+HEIGHT C DO 700 I=1,BNS C DO 650 d=1,BNL C NOD=LAST+d RC(N0D)=BWL*(J-1) ZC(NOD)=BHS*I+PHT 650 CONTINUE C LAST=LAST+BNL+1 RC(LAST)=SEEDW ZC(LAST)=BHS*I+PHT C DO 680 J=1,BNL C NEL0=2*J-1+LASTEL NODE(NE LO, 1 ) = LAST-2*BNL+J-2 NODE(NELO,2) = LAST-BNL+J N0DE(NEL0,3)=N0DE(NEL0,2)-1 C NELP=2*d+LASTEL NODE(NELP, 1)=N0DE(NE LO, 1) N0DE(NELP,2)=NODE(NELP,1)+1 NODE(NELP,3)=N0DE(NEL0,2) C 680 CONTINUE C LASTEL=LASTEL+2*BNL C 70Q CONTINUE C " NN^LAST NE=LASTEL C C DETERMINE SEGMENTS WHERE CONVECTION OCCURS C DO 800 1=1,NBC HV(I)=0.6 TINFV(I)=0.0 NV1(I)=(NBCL+1)*I+NNINT NV2(I)=(NBCL+1)*(1+1)+NNINT 800 CONTINUE 393 DO 810 1=1,BNS I$=I+NBC HV(I$)=0.3 TINFV(I$)=0.0 NV1(I$)=LASTN(LL)+(BNL+1)*(I-1) NV2(I$)=NV1(I$)+BNL+1 810 CONTINUE C DO 820 1=1,CBN HT(I)=0.S TINFT(I)=0.0 NT 1(I) = LASTN(I) d = I+1 NT2(I)=LASTN(d) 820 CONTINUE C DO 830 1=1,BNL HH(I)=0.0 TINFH(I)=0.0 NH1(I)=NN-I+1 NH2(I)=NN-I 830 CONTINUE C NNHCV=NBC+BNS NNHCT = CBN NNHCH=BNL C C PRINT THE DATA IN FORMAT TO BE READ BY FLEMT AND PFLEMS C WRITE(6,1005) NN,NE WRITE(G,100G) NNHCV.NNHCH,NNHCT 1005 F0RMAT(5X.'NN= '.I4,5X,'NE= ',14,//) 1006 FORMAT(5X,'NNHCV=',13,'NNHCH=',13,'NNHCT=',13,/) DO 900 1=1,NN ZC(I ) = ZC(I) + ZINT WRITE(6, 1020)1,RC(I ) , ZC( I ) 900 CONTINUE C DO 910 K=1,NE WRITE(6, 1025)K,NODE(K, 1 ) ,N0DE(K,2),N0DE(K,3) 910 CONTINUE C NNCL=NBCL+1 T= 1 .0 DO 920 1 = 1 ,NNCL WRITE(6, 1030)1,T 920 CONTINUE C DO 930 1=1,NNHCV WRITE(6, 1040)NV 1(1),NV2(I),HV(I),TINFV( I ) 930 CONTINUE C DO 940 1 = 1 .NNHCT WRITE(6,1040)NT1(1),NT2(I),HT(I),TINFT(I) 940 CONTINUE C DO 950 1=1,NNHCH WRITE(6,1040)NH1(I),NH2(I),HH(I),TINFH(I) 950 CONTINUE C C FORMAT STATEMENTS C 1010 FORMAT(2(2X,F10.5)) 1015 FORMAT(3(2X,F10.5)) 1020 FORMAT(2X, I4,2(2X,F10.5) ) 1025 F0RMAT(4(2X,14)) 1030 F0RMAT(2X,I3.2X.F10.5) 1040 F0RMAT(2(2X.I4).2(2X,F10.5)) STOP END 394 XI.2 Finite Linear Element Program for the Temperature Calculations during Growth CTHIS IS A PROGRAM TO CALCULATE THE TEMPERATURE DISTRIBUTION C IN LEC-GAAS DURING GROWTH USING A FINITE LINEAR ELEMENT C METHOD (SECOND VERSION). C C DIMENSION IP(375),RES(375) INTEGERM LLB , LUB , IP, ITER , NHRS REAL*4 RES,EPS REAL *8 TB.DRATIO INTEGERS LHB,NRHS.JEXP,NSCALE DIMENSION TB(2) INTEGER*4 NN,NDIMA,NSOL,NDIMBX,IPERM,NDIMT REAL*8 TM,RINF,T,TT,DET REAL*8 DTA , RBAR,A 1,A2,A3,B1,B2,B3 REA L *8 C1 ,C2,C3,VK,TM$,CONST,DEL2 REAL*8 BLV,CONSTV,CINFV,BLT.CONSTT REAL*8 CINVT,BLH.CONSTH,CINFH,RBARH DIMENSION TM(111,111),T(111),RINF(111),IPERM(222),TT(11 1 , 1 11) DIMENSION RC(500),ZC(500).NODE(900,3) DIMENSION NTS(50),HV(60),TINFV(60),HH(60),TINFH(60) DIMENSION BLV(60),NV1(60),NV2(60) DIMENSION BLT(40),NT 1(40),NT2(40),HT(40),TINFT(40) DIMENSION BLH(40),NH1(40),NH2(40),RBARH(40) DIMENSION TM$(3,3),N(3),RC$(3),ZC$(3) DIMENSION TG(50.50),TGR(111),CVAL(20) ,I0P(8),V0P(8) REAL TG,CVAL,VOP,TMIN,DTEM,DX,DY,C,XSIZE,YSIZE INTEGER NX,NY,NRNG.IDIMX,NC,IOP REAL BORAXH,HBORAX,HARGON REAL TINFAR,TINFBO,ZBAR,TMBOR REAL TBOR,CA2.CA3,CA4,CB2,CBS,CB4 C C C C READ THE NUMBER OF NODES AND ELEMENTS OF THE SYSTEM C C READ(5,1005)NN,NE C C INITIALIZE PARAMETERS C C DO 10 1=1,NN RINF(I) = 0.0 RC(I) = 0.0 ZC(I) = 0.0 T(I) = O.O DO 10 J = 1,NN TM(I,J ) = 0.0 10 CONTINUE C C READ THE NUMBER OF NODES WITH SPECIFIED TEMPERATURES C AND THE NUMBER OF BOUNDARY SEGMENTS.VERTICAL AND NON C VERTICAL,WHERE HEAT CONVECTION OCCUR. C C RE AD ( 5 , 1010 )NNST , NNHCV , NNHCH , NNHCT C C READ GROWTH PARAMETERS:RAD IUS,CONDUCTIVITY,VELOCITY C AND LENGTH. C READ(5,1015)R0,TK,V,XL,DTA C READ(5,1018) CA,SW,SH,BORAXH READ(5.1019) CB2.CB3,CB4.CA2,CA3,CA4 C WRITE(6.1019) CB2,CB3.CB4.CA2,CA3,CA4 C C READ NUMBER OF NODE VS. NODAL COORDINATES C DO 100 d = 1,NN READ(5 , 1020)1.RC(I ) ,ZC(I ) 100 CONTINUE C C READ ELEMENT NUMBER VS. NODE NUMBERS C DO 105 I = 1.NE READ(5,1O25)d.N0DE(d.1),N0DE(J,2),N0DE(d,3) 105 CONTINUE C C READ NODE NUMBER VS. SPECIFIED TEMPERATURE C DO 110 I = 1,NNST READ(5.1030) NT,TNT NTS(I ) = NT T(NTS(I)) = TNT 110 CONTINUE C C READ PAIRS OF NODE NUMBERS DEFINING BOUNDARY SEGMENT C WHERE CONVECTION OCCURS,CALCULATE THE CORRESPONDING C CHTC AND AMBIENT TEMPERATURE. C C TMBOR=1238.0 TMBOR=1.-(1238.-TMBOR)/DTA TBOR = TINFBO(DTA,TMBOR,CB2.CB3,CB4,3.000) DO 140 I = 1 ,NNHCV REA0(5,1O4O) NV1(1),NV2(I) ZBAR=(ZC(NV1(I))+ZC(NV2(I)))*R0/2. IF(ZBAR.LT.3.OOO) TINFV(I)=TINFBO(DTA,TMBOR,CB2.CBS,CB4,ZBAR) IF(ZBAR.GT.3.000.OR.ZBAR.EO.3.000) TINFV(I)=TINFAR(DTA,TBOR,CA2. 1CA3.CA4.ZBAR) C IF(ZBAR . LT . BORAXH.OR.ZBAR.EO.BORAXH) HV(I)=HBORAX(DTA,TINFV(I)) IF(ZBAR.GT.BORAXH) HV(I)=HARG0N(DTA,TINFV(I)) C 140 CONTINUE C C DO 145 d=1,NNHCT READ(5,1040)NT1(d),NT2(d) ZBAR=(ZC(NT1(d))+ZC(NT2(d)))*R0/2. C IF(ZBAR.LT.3.000) TINFT(d)=TINFBO(DTA,TMBOR,CB2,CBS,CB4,ZBAR) IF(ZBAR.GT.3.000.0R.ZBAR.EO.S.OOO) TINFT(d)=TINFAR(DTA, 1 TBOR.CA2.CA3.CA4.ZBAR) C IF(ZBAR.LT.BORAXH) HT(d)=HBORAX(DTA.TINFT ( d)) IF(ZBAR GT.BORAXH) HT ( d)=HARGON(DTA,TINFT(d ) ) C 145 CONTINUE DO 150 d = 1,NNHCH REA0(5,1O4O) NH1(d),NH2(d) ZBAR=ZC(NH1(d))*R0 TINFH(d)=TINFAR(DTA,TBOR,CA2,CA3,CA4,ZBAR) HH(d ) =HARGON(DT A,TINFH(d)) C 150 CONTINUE C C WRITE HEADLINES C 155 WRITE(6,2010) WRITE(6,2020) RO.V.XL WRITE(6,2030) NN,NE 396 WRITE(6,2035) CA.SW.SH WRITE(6,2038) DTA,BORAXH C C C C PRINT SPECIFIED TEMPERATURES C 175 WRITE(G,2080) DO 180 I = 1 ,NNST TSP= 1238. WRITE(6,2090) I,NTS(I),TSP 180 CONTINUE C C PRINT BOUNDARY SEGMENT NODES,HTCH AND HTCV,AND C AMBIENT TEMPERATURE. C WRITE(6,2100) DO 190 I = 1 , NNHCV-TINF=1238-(1. -TINFV(I))*DTA WRITE(6,2110) I.NV1(I),NV2(I),HV(I ) ,TINF 190 CONTINUE C IF(NNHCT.EO.1) GO TO 196 WRITE(6,2105) DO 195 1=1,NNHCT TINF =1238-(1.-TINFT(I))*DTA WRITE(6,2 110)1,NT1(I),NT2(I),HT(I),TINF 195 CONTINUE C C 196 IF(NNHCV.EO.1) GO TO 250 WRITE(6,2 120) DO 200 1=1,NNHCH TINF = 1 238-( 1 .-TINFH(I))*DTA WRITE(6,2110) I.NH1(I),NH2(I),HH(I),TINF 200 CONTINUE C C CYCLE FOR EACH ELEMENT, K , AND FORM THE INFLUENCE C MATRIX OF THE SYSTEM C C 250 DO 400 K = 1,NE N1 = NODE(K,1) N2 = N0DE(K,2) N3 = NODE(K,3) RC$( 1 ) = RC(NODE(K, 1 ) ) RC$(2) = RC(N0DE(K,2)) RC$(3) = RC(N0DE(K,3)) ZC$( 1 ) = ZC(NODE(K, 1)) ZC$(2) = ZC(N0DE(K,2)) ZC$(3) = ZC(N0DE(K,3)) RBAR = (RC$(1) + RC$(2) + RC$(3))/3. A1 = RC$(2)*ZC$(3) - RC$(3)*ZC$(2) A2 = RC$(3)*ZC$(1 ) - RC$(1)*ZC$(3) A3 = RC$(1)*ZC$(2) - RC$(2)*ZC$( 1 ) B1 = ZC$(2) - ZC$(3) B2 = ZC$(3) - ZC$(1) B3 = ZC$( 1 ) - ZC$(2) C1 = RC$(3) - RC$(2) C2 = RC$( 1 ) - RC$(3) C3 = RC$(2) - RC$(1) DEL2= ABS(RC$(1)*(ZC$(2)-ZC$(3)) + RC$(2)*(ZC$(3 )-ZC$(1 ) ) + RC$(3) 1*(ZC$(1)-ZC$(2))) C C CALCULATE THE ELEMENTS OF THE MATRIX FOR EACH ELEMENT C VK = (3. 14 15927*V*R0)/(12.0*TK) CONST = (3. 1415927*RBAR)/DEL2 397 c TM$(1, , 1) = = (B1 ••2 + C1 ••2) *CONST + (2.•RC$( 1 ) + RC$(2) + RC$(3) )*B 1 1 *VK TM$(1, ,2) = = (B1 *B2 + C1 *C2) •CONST + (RC$(1) + 2.4RC$(2) + RC$(3))*B 12*VK TM$(2, , 1) = = (B1 *B2 + C1 *C2) •CONST + (2 . *RC%( 1 ) + RC$(2) + RC$(3))*B 1 1 *VK TM$(2, ,2) = = (B2 **2 C2 **2) •CONST + (RC$(1) + 2.*RC$(2) + RC$(3) )^B 12*VK TM$( 1 , 3) = = (Bl *B3 + CI •C3) •CONST + (RC$(1) + RC$(2) + 2 •RC$(3))^B 13*VK TM$(3, , 1 ) = = (B1 *B3 + C1 *C3) •CONST + (2 . *RC$( 1 ) + RC$(2) + RC$(3))*B 1 1*VK TM$(2, .3) = = (B2 *B3 + C2 *C3 ) •CONST + (RC$( 1 ) + RC$(2) + 2 •RC$(3))^B 13*VK TM$(3, 2) = = (B2 *B3 + C2 *C3) •CONST + (RC$(1) + 2.*RC$(2) + RC$(3) )*B 12*VK TM$(3, .3) = = (B3 •*2 + C3 •*2) •CONST + (RC$(1) + RC$(2) + 2 •RC$(3) )*B 13*VK C C ASSEMBLE SYSTEM MATRIX WITHOUT REGARDING BOUNDARY CONDITIONS C N(1) = N1 N(2) = N2 N(3) = N3 DO 300 1$ = 1,3 I = N(I$) DO 300 J$ = 1.3 J = N(d$) TM(I.d) = TM(I,«J) + TM$(I$,J$) 300 CONTINUE C CCYCLE FOR NEXT ELEMENT C 400 CONTINUE C C ACCOUNT FOR CONVECTION AT BOUNDARY,FORM RINF MATRIX C DO 420 I = 1,NNHCV BLV(I) = ABS(ZC(NV1(I)) - ZC(NV2(I))) CONSTV = 6. 1853832*BLV(I )*HV(I)+R0+RC(NV1(I)) TM(NV1(1),NV1(1)) = TM(NV1(1),NV1(1)) + CONSTV/3. TM(NV1(1),NV2(I)) = TM(NV1(1),NV2(I)) + CONSTV/6. TM(NV2(I),NV1(I)) = TM(NV2(I),NV1(I)) + CONSTV/6. TM(NV2(I),NV2(I ) ) = TM(NV2(I),NV2(I)) + CONSTV/3. C CINFV = 3.1416 4HV(I)4TINFV(I)•RO^RC(NV1(I))4BLV(I) RINF(NV1(I ) ) = RINF(NV1(I)) + CINFV RINF(NV2(I)) = RINF(NV2(I)) + CINFV 420 CONTINUE C IF(NNHCT.EO.1) GO TO 435 DO 430 1=1,NNHCT BLT(I)=ABS(SORT((RC(NT 1 (I ) )-RC(NT2(I)))*•2+(ZC(NT 1(I))-ZC(NT2(I)) 1 )**2) ) RBAR = (RC(NT 1(I)) + RC(NT2(I)))/2. C0NSTT=3.1416AHT(I)•R0iBLT(I)/6. C TM(NT 1(I).NT 1(I))=TM(NT 1(I),NT 1(I))+CONSTT•(3.*RC(NT 1(I)) + 1RC(NT2(I))) C TM(NT 1(I),NT2(I))=TM(NT1(I),NT2(I))+C0NSTT*2.+RBAR C TM(NT2(I),NT 1(I))=TM(NT2(I),NT 1(I))+C0NSTT^2.•RBAR 398 c TM(NT2(I),NT2(I))=TM(NT2(I),NT2(I)) + CONSTT*(RC(NT 1(I ) ) + 13.* RC(NT 2(1 ) ) ) C CINFT=3.1416*HT(I)*RO*BLT(I)*TINFT(I)/3. C RINF(NT1(I))=RINF(NT1(I))+ CINFT*(RC(NT 1(I)) + 2.*RC(NT2(I))) C RINF(NT2(I ) ) = RINF(NT2(I ))+CINFT*(2.*RC(NT2(I))+RC(NT 1(I))) C 430 CONTINUE 435 IF(NNHCH.EO.1) GO TO 445 DO 440 d = 1 ,NNHCH BLH(d) = SQRT((RC(NH1(d)) - RC(NH2(d)))* *2 + (ZC(NH1(d)) - ZC(NH2( 1d)))**2) RBARH(d) = (RC(NH1(d)) + RC(NH2(d)))/2. CONSTH = 6.2832*R0*HH(d)*BLH(d)*RBARH(d) TM(NH1(d),NH1(d)) = TM(NH1(d),NH1(d)) + CONSTH/3. TM(NH1(d),NH2(d)) = TM(NH1(d),NH2(d)) + C0N5TH/6. TM(NH2(d),NH1(d)) = TM(NH2(d).NH1(d)) + CONSTH/6. TM(NH2(d),NH2(d)) = TM(NH2(d),NH2(d)) + CONSTH/3. CINF = 3.1416 *HH(d)*TINFH(d)*RO*RBARH(d)*BLH(d) RINF(NH1(d)) = RINF(NH1(d)) + CINF RINF(NH2(d)) = RINF(NH2(d)) + CINF 440 CONTINUE C C MODIFY TM AND RINF TO ACCOUNT FOR NODES WITH SPECIFIED C TEMPERATURE C 445 DO 450 I = 1 ,NNST DO 448 d=1.NN RINF(d)=RINF(d)-TM(d,NTS(I)) TM(NTS(I) ,d)=0.0 TM(d,NTS(I))=0.0 448 CONTINUE RINF(NTS(I))=1. TM(NTS(I ) ,NTS(I)) = 1. 450 CONTINUE C C C CALL SOUBROUTINE FSLE TO SOLVE THE LINEAR SYSTEM C OF EQUATIONS C 458 NDIMA = NN NSOL = 1 NDIMBX = NN NDIMT = NN CALL SLE(NN,NDIMA,TM,NSOL,NDIMBX,RINF,T,IPERM,NDIMT,TT,DET,dEXP) GO TO 475 C C 4G0 LHB=21 IdMAX=(LHB-1)*NN+NN-(LHB-1) DO 465 1=1,IdMAX TB(I)=0.O 465 CONTINUE DRAT 10= 1 .E-5 NRHS=1 NSCALE=0 DO 470 d=1,NN DO 470 I$=1,LHB I=d+I$- 1 IF(I.GT.NN) GO TO 470 Id=(LHB-1)*(d-1)+I TB(Id)=TM(I,d) 470 CONTINUE C C CALL DFBAND C CALL DFBAND(TB,RINF,NN,LHB,NRHS.DRAT 10 , 1 DET,dEXP,NSCALE) u 475 WRITE(6,2500) DET,dEXP C C CPRINT OUTPUT C WRITE(G,2200) DO 500 I = 1,NN TGR(I)=1.238-(1.-T(I))*DTA/1000. WRITE(6,2300) I,T(I),TGR(I) 500 CONTINUE C C CALCULATE NEW GRID C NX = 50 NY = 50 IDIMX=NX C TMIN=TGR(1) C DO 550 I=1,NN 550 IF(TMIN.GT.TGR(I)) TMIN=TGR(I) C WRITE(6,2400) TMIN C XMIN=0.0 YMIN=0.0 XMAX=10. YMAX=10. C DX= (XMAX-XMIN)/(NX-1) DY=(YMAX-YMIN)/(NY-1) C NRNG=2 C=10. 0PT=O.O C DO 600 I=1,NN RC ( I ) = ( 1.-RC(I))*5.0 ZC(I) = ZC(I )*5 .0 600 CONTINUE C CALL CGRID1(TG,IDI MX,NX,NY,XMIN,YMIN, 1 DX,DY,RC,ZC.TGR.NN,C,NRNG,OPT) C C PLOT GRID C XSIZE=10. YSIZE=10. NC=12 I0P(1)=1 I0P(2)=1 I0P(3)=0 I0P(4)=1 I0P(5)=1 V0P(5)=3. I0P(G)=1 I0P(7)=0 I0P(8)=1 V0P(8) = . 12 C DTEM=(1.238-TMIN)/(NC-1) C DO 700 1=1,NC 700 CVAL(I)=1.238-DTEM*(1-1) C CALL CONTUR(XSIZE,YSIZE,TG.IDIMX.NX.NY, 1 CVAL.NC,IOP.VOP) C CALL PLOTND C C C FORMAT STATEMENTS C 1005 FORMAT(2(2X,13 ) ) 1010 F0RMAT(4(2X,I3)) 1015 F0RMAT(2X,F5.2,2X,F7.4,2X,F7.5,2(2X,F7.3)) 1018 F0RMAT(4F12.5) 1019 F0RMAT(6(2X,F10.5)) 1020 FORMAT(2X,I4,2(2X,F10.5)) 1025 F0RMAT(4(2X,14) ) 1030 F0RMAT(2X,13,2X,F10.5) 1040 FORMAT(2(2X,14)) 2010 FORMAT(1H1,2X,' A FLEM SOLUTION FOR THE TEMPERATURE FIELD IN LEC-G 1AAS CRYSTALS',//) 2020 F0RMAT(5X,'RADIUS (CM) = ', F7 . 5.//,5X,'VELOCITY (CM/SEC) = ',F7.5, 1//5X,'LENGTH (CM) = ',F7.3.//) 2030 FORMAT(5X. 'NO. OF NODES = ' ,I 3,//,5X, 'NO. OF ELEMENTS = ',13,//) 2035 F0RMAT(5X, 'CONE ANGLE =' ,F 1 2 . 5, 'DEG. ' ./, 1 5X,'SEED WIDTH = ' ,F 12 . 5, 'CM',/, 2 5X.'SEED HEIGHT = ' .F 1 2 . 5, 'CM' ,/) 2038 F0RMAT(5X, 'TMP-TINF =' ,F 1 2 . 5, 'DEG. C',/, 1 5X, 'BORAX HEIGHT =' .F 1 2.5. ' CM' ,/) 2040 FORMAT( 10X, 'NODES VS. NODAL COORDINATES',//,6XI', 13X .'R(I )', 13X, l'Z(I)' .//) 2050 FORMAT(5X,I3,2(7X,F10.5)) 2060 FORMAT(' ',///,15X,'SYSTEM TOPOLOGY',//,'ELEMENT NUMBER',11X,'NODE 1 NUMBERS' ,// ) 2070 F0RMAT(5X , 13, 10X,3(5X,13)) 2080 FORMAT( ' ',///,5X,'NODES WITH SPECIFIED TEMPERATURES',//.7X.'I',4X 1,'NODE',6X,'TEMPERATURE',//) 2090 FORMAT(2X,2(4X,I3),5X,F15.7) 2100 FORMAT( ' './/,'BOUNDARY SEGMENTS WHERE CONVECTIVE HEAT FLUX OCCUR' 1,//,5X , ' I ' ,6X , 'NODE NUMBER PAIR ' ,4X, 'H' ,8X, 'TINF' .//,20X, 'VERTICAL 2 BOUNDARY' ,/ ) 2105 FORMAT(' './/,2X,'TILTED BOUNDARY',//) 2110 FORMAT* 2X,3(3X,13),2(8X,F 10 . 5 ) ) 2120 FORMAT( ' ' ,/,20X. 'NON-VERT ICAL BOUNDARY',/) 2200 FORMAT( ' ' ,//,5X, 'NODE NUMBER', 10X,'T(I)',//) 2300 FORMAT( 10X , 13.6X,2(F15.4 ) ) 2400 FORMAT(//, 10X , 'TMIN= '.F15.6) 2500 FORMAT( 'DETERMINANT IS ' ,G16.7, '10**' ,I 20.//) STOP END C C C FUNCTIONS TO CALCULATE TINF C C REAL FUNCTION TINFBO(DTA,TMBOR,CB2,CB3,CB4,ZZ) REAL DTA,CB2.CB3,CB4.ZZ,TMBOR C TINFBO=TMBOR+(CB2*(ZZ-O.000)+CB3*(ZZ-O.000)**2+ 1CB4*(ZZ-0.000)**3)/DTA C RETURN END 401 c c REAL FUNCTION TINFAR(DTA,TBOR,CA2,CA3,CA4,ZZ ) REAL DTA,TBOR,CA2,CA3,CA4,ZZ C TINFAR=TB0R+(CA2*(ZZ-3.0OO)+CA3*(ZZ-3.OOO)**2+ 1CA4*(ZZ-3.000)**3)/DTA C RETURN END C C C FUNCTIONS TO CALCULATE HTC C C REAL FUNCTION HBORAX(DTA,TE) REAL DTA,TE C HBORAX=0.8*EXP(1 .041E-3*(1 1 .-DTA *(1.-TE ) ) -13 . 267 1E-6*( 1 1.-DTA*(1 .-TE) )**2 ) C RETURN END C C REAL FUNCTION HARGON(DTA,TE ) REAL DTA,TE C HARGON=0.66*EXP(3.2319E-3*(11.-DTA *(1.-TE))) C RETURN END 402 XI.3 Finite Linear Element Program for the Stress Calculations c C C THIS PROGRAM CALCULATES THE DISPLACEMENTS AND STRESSES C IN LEC_GAAS CRYSTALS DURING GROWTH USING A FLEM METHOD C C REAL DVMS.XIM.XI,XCN REAL*4 STR,E$ DIMENSION STR(6),E$(3) I NT EGER*4 I ERROR,NVEC REAL *8 E.AB DIMENSION AB(21),E(6) INTEGER *4 NN.NEQ.LH.U INTEGER*4 JS . JSJS . US , JSJ I NT EGER* 4 NSOL,I PERM.NSCALE,LHB INTEGER*4 LLB,LUB,IP,JEXP,ITER INTEGER*4 NHRS RE AL *8 EPS.RES REAL*8 DAA,DRATIO DIMENSION DAA(40000) REAL*8 DSS.DDET DIMENSION DSS(992) DIMENSION SIGRB$(3),SIGTB$(3),SIGZB$(3),TAUB$(3),SSP$(G) DIMENSION DSK$(6,6),A(3),B(3),C(3),DSS$(6) DIMENSION T$(3),N(3),RC$(3),ZC$(3) DIMENSION RC(600),ZC(600),NODE(1000,3),T(GOO),NSRD(100),NSAD(100) DIMENSION U(600),V(600),NSAS(100) DIMENSION U$(3) ,V$(3),SIGR$(3),SIGTH$(3),SIGZ$(3),TAURZ$(3),XBAR(3 1 ) DIMENSION SIGRP(600),SIGTHP(600),SIGZP(600),TAURZP(600),NEPN(600) DIMENSION DEL2P(1000) DIMENSION NSS(300) DIMENSION W(7),XL(3,7) REAL*8 DSK$1 ,DSK$,DSS$ REAL*8 FUNC,C0NS,ALPHA1,ALPHA2,BETA 1,BETA2 REAL+8 EPR,EPZ,GRZ,TBAR,U,V REAL*8 XL,W,RBAR,DEL2,DEL2P,A,B,C C DIMENSION VMS(1GG),VONMS(50,50),TEMP(50,50) DIMENSION CVAL(10),I0P(8),V0P(8) REAL VONMS,TEMP,XMIN,XMAX,YMIN,YMAX REAL DX,DY,VOP,CVAL,XSIZE,YSIZE,S INTEGER IDIMX,IOP,NX,NY,NC,NRNG C C READ THE NUMBER OF NODES AND NODES WITH SPECIFIED DISPLACEMENT C READ(5,1005)NN,NE,NNSRD,NNSAD C C INITIALIZE VARIABLES C NEO = 2*NN C C LHB=26 LH=LHB-1 C UMAX = LH*(NEQ-1)+NEO C DO 8 1 = 1 , UMAX DAA(I)=0.0 8 CONTINUE C DO 10 1 = 1 ,NEO DSS(I) = 0.0 C 10 CONTINUE DO 20 1 = 1 ,NN U(I) = 0.0 V(I) = 0.0 RC( I ) = 0..0 ZC(I) = 0.0 T(I) = 0.0 SIGRP(I) = 0.0 SIGTHP(I) = 0.0 SIGZP(I ) =0.0 TAURZP(I ) = 0.0 DEL2P(I)=0.0 20 CONTINUE C C READ CONSTANT AND COEFFICIENTS C READ(5,1010) XNU READ(5,1015) RO.VEL.XLE.DTA C DO 100 J = 1,NN READ(5,1020) I , RC(I),ZC(I) 100 CONTINUE C C READ SYSTEM TOPOLOGY C DO 105 I = 1,NE READ(5,1025)J,NODE(J, 1),NODE(J,2),NODE(J,3 ) 105 CONTINUE C C C READ NODE NUMBERS WITH SPECIFIED DISPLACEMENT C NNSRD=0 DO 110 I = 1,NN IF(RC(I ) .GT.0.0) GO TO 110 NNSRD=NNSRD+1 NSRD(NNSRD ) = I WRITE!6,1030) NSRD(NNSRD) 110 CONTINUE C C C READ'TEMPERATURES c DO 120 <J = 1 , NN READ(5,1040)1,T(I) 120 CONTINUE C C PRINT DATA AND HEADLINES C WRITE(6,2010) WRITE(6.2020)R0,VEL.XLE WRITE(6,2030)XNU WRITE(6.2035)NN,NE.NNSRD C C C PRINT HEADLINES FOR DISPLACEMENTS C WRITE(6,2100) C C FORM MATRIX FOR INITIAL STRAIN AND STIFFNESS MATRIX C C C GIVE VALUES TO WEIGHT FUNCTIONS AND NATURAL C C COORDINATES FOR NUMERICAL INTEGRATION OF C C OUINTIC ORDER C 404 ALPHA 1 = .059715871789770 BETA 1 =.470142064 1051 15 ALPHA2= . 797426985353087 BE TA2 = . 101286507323456 W( 1)= .225 W(2)=.132394 152788506 W(3)=W(2) W(4)=W(3) W(5)=. 125939180544827 W(6)=W(5) W(7)=W(6) XL( 1, 1 ) = 1 ./3. XL(1,2)=ALPHA1 XL(1,3)=BETA1 XL(1,4)=BETA1 XL(1,5)=ALPHA2 XL(1.6)=BETA2 XL(1.7)=BETA2 XL(2 , XL(2, XL(2, XL( 2 , XL( 2 , XL( 2 , XL( 2 , 1 ) = 1 ./3• 2) =BETA1 3) =ALPHA1 4) =BETA 1 5) =BETA2 6) =ALPHA2 7) =BETA2 140 DO 140 1=1 XL(3,I)=1.• CONTIMUE DO 400 K=1 7 (XL(1, NE I)+XL(2,I)) N( 1 ) N(2) N(3) NODE(K,1) NODE(K,2) NODE(K,3) TBAR=(T(NODE(K,1))+T(NODE(K, T$(1) = TBAR T$(2) = TBAR 2))+T(NODE(K,3)))/3. T$(3) TBAR RC$( 1 ) = RC(NODE(K, 1)) RC$(2) = RC(N0DE(K,2)) RC$(3) = RC(N0DE(K,3)) ZC$( 1) = ZC(N0DE(K, 1 ) ) ZC$(2) = ZC(N0DE(K,2) ) ZC$(3) = ZC(N0DE(K,3)) A(1) = RC$(2)*ZC$(3) A(2) = RC$(3)*ZC$(1) A(3) = RC$( 1 )*ZC$(2) RC$(3)*ZC$(2) RC$(1)*ZC$(3) RC$(2)*ZC$(1) B(1) B(2) B(3) ZC$(2) ZC$(3) ZC$( 1 ) ZC$(3) ZC$(1) ZC$(2) C(1) = RC$(3)- RC$(2) C(2) = RC$( 1 ) - RC$(3) C(3) = RC$(2) - RC$( 1 ) RBAR = (RC$(1) + RC$(2) + RC$(3))/3. ZBAR = (ZC$(1) + ZC$(2) + ZC$(3))/3. DEL2 = ABS(RC$(1)*(ZC$(2) - ZC$(3)) + RC$(2)*(ZC$(3) 1RC$(3)*(ZC$( 1 ) - ZC$(2) ) ) ZC$(1))+ 405 c DO 200 IS= 1 .3 C C FORM MATRIX FOR INITIAL STRAIN IN ELEMENT K C K$ = 2*1$ - 1 DSS$(K$)=T$(I$)*(B(I$)*RBAR+DEL2/3.)/2. C K$ = 2*1$ DSS$(K$) = T$(I$)*RBAR*C(I$)/2. C C FORM STIFFNESS MATRIX FOR ELEMENT K C DO 200 J$ = 1,3 C C INTEGRATE NUMERICALLY TERM IN L(I$)*L(J$ )/R0 C FUNC=0.0 C DO 195 1=1,7 FUNC=FUNC+W(I)*XL(I$,I)*XL(J$,I)/ 1 (RC$( 1 )*XL( 1 ,I )+RC$(2)*XL(2,I) + RC$(3)*XL( 3, I)) 195 CONTINUE C IF(I$.EO.d$) GO TO 198 PIN=(1.-XNUj*DEL2/(24.*RBAR) GO TO 199 198 PIN=(1.-XNU)*DEL2/(12.*RBAR) C C 199 DSK$1=((1.-XNU)*B(I$)*B(J$)+(.5-XNU)*C(I$)*C(J$))*RBAR/(2.*DEL2) C K$ = 2*I$- 1 L$=2*J$-1 C DSK$(K$,L$)=DSK$1+XNU*(B(I$)+B(d$))/6.+FUNC*DEL2*(1-XNU)/2. C K$=2*I$ C DSK$(K$,L$)=(XNU*B(J$)*C(I$)+(.5-XNU)*B(1$)*C(J$))* 1RBAR/(2.*DEL2)+XNU*C(I$)/6. C L$=2*J$ C DSK$(K$, L$) = (( 1 .-XNU)*C(I$)*C(d$) + ( .5-XNU)*B(I$)*B(J$))* 1RBAR/(2.*DEL2) C K$ = 2*I$- 1 C DSK$(K$,L$)=(XNU*B(I$)*C(U$)+(.5-XNU)*B(0$)*C(1$))* 1RBAR/(2.*DEL2)+XNU*C(J$)/6. C C C CYCLE FOR NEXT NODE C 200 CONTINUE C GO TO 262 C C C CALCULATE EINGENVALUES OF THE C STIFFNESS ELEMENT MATRIX C C WRITE(6,2510)K C DO 250 I$= 1 ,6 DO 250 J$=1,1$ I$J$=I$*(I$-1)/2+d$ AB(I$J$)=DSK$(I$,J$) 250 CONTINUE C 406 NVEC=0 NEQ$=6 CALL D5YMAL(AB,NE0$,E,I ERROR,NVEC) C DO 260 I$= 1 .6 WRITE(6,261) I$.E(I$) 260 CONTINUE C 261 F0RMAT(I3,F12.8) WRITE*6,2510) I ERROR C C C C ASSEMBLE MATRICES WITHOUT REGARDING B.C. C 262 DO 300 I$ = 1 . 3 C I = N(1$) C M$ = 2*1$ - 1 M = 2*1 - 1 DSS(M) = DSS(M) + DSS$(M$) C M$ = 2*1$ M = 2*1 DSS(M) = DSS(M) + DSS$(M$) C C DO 300 d$ = 1,3 C J = N(J$) C M$ = 2*1$ - 1 M = 2*1 - 1 N$ = 2*d$ - 1 N1 = 2*J - 1 IF(N1.GT.M) GO TO 280 C Id=LH*(N1-1)+M C DAA(Id)= DA A (1d)+DSK$(M$,N$) C C 280 N$ = 2*d$ N1 = 2*d IF(N1.GT.M) GO TO 285 C IJ = LH*(N1- 1 )+M C DAA(Id)=DAA(Id)+DSK$(M$,N$) C C 285 M$ = 2*1$ M = 2*1 IF(N1.GT.M) GO TO 290 C IJ=LH*(N1-1)+M C DAA(Id)=DAA(Id)+DSK$(M$,N$) C C 290 N$ = 2*d$ - 1 N1 = 2*d - 1 I F (N1.GT.M) GO TO 300 C Id = LH*(N1-1 )+M C DAA(Id)=DAA(Id)+DSK$(M$,N$) C • C C ASSEMBLE ELEMENTS OF THE NEXT NODE IN ELEMENT K C 300 CONTINUE c C CYCLE FOR NEXT ELEMENT C 400 CONTINUE C C MODIFY SK AND SS TO ACCOUNT FOR C C NODES WITH SPECIFIED DISPLACEMENT C C C FOR AXIAL NODES NO RADIAL C , DISPLACEMENT C DO 500 d$ =1,NNSRD dS=2*NSRD(d$)-1 dSdS=LH*(JS-1) + JS DAA(JSJS)=1 .0 DSS(JS)=0.0 DO 490 I$=1,LH I=JS+I$ IJS=LH*(JS-1)+I DAA(I JS)=0.O C 485 IF(I$.GT.JS.OR.I$.EQ.JS) GO TO 490 J=dS-I$ dSd=LH*(d-1)+JS DAA(dSd)=0.0 490 CONTINUE 500 CONTINUE C C C FOR NODE 1,N0 AXIAL C DISPLACEMENT C JS = 2 dSJS=LH*(JS-1) + JS DAA(JS JS ) = 1 .O DSS(JS)=0.0 C DO 505 I$=1,LH I=JS+I$ IdS=LH*(dS-1)+I DAA(IdS)=0.0 IF(1$.GT.dS.OR.1$.EQ.JS) GO TO 505 J=JS-I$ JS«J = LH*(d-1 ) + JS DAA(dSd)=0.0 505 CONTINUE C C DRATIO=1.E-5 NRHS=1 NSCALE=0 C C SOLVE THE LINEAR SYSTEM C C CALLING DFBAND C C C CALL DFBAND(DAA.DSS.NEO,LHB,NRHS,DRATIO,DDET,dEXP,NSCALE ) C WRITE(6,2005) DDET.dEXP C C C PRINT DISPLACEMENTS C C WRITE(6,2115) DO SOO I = 1,NN d = 2*1 - 1 U(I) = DSS(d) K = 2*1 V( I )=DSS(K) 408 c WRITE(6.2110)1,U(I),V(I) 600 CONTINUE C C C CALCULATE STRESSES AT ALL NODES IN EACH ELEMENT C C DO 700 I = 1,NE C N( 1)=NODE( 1,1) ' N(2)=N0DE(I,2) N(3)=N0DE(I.3) C RC$( 1 ) = RC(NODE(I . 1 ) ) RC$(2) = RC(NODE(1,2)) RC$(3) = RC(NODE(1,3)) C ZC$( 1 ) = ZC(NODE(I,1) ) ZC$(2) = ZC(NODE(1,2)) ZC$(3) = ZC(N0DE(I,3)) C C TBAR=(T( NODE (I , 1 ) ) + T (NODE ( I , 2 ) ) + T (NODE ( I . 3 ) ) )/3. C C C C T$( 1 ) = = TBAR T$(2) = = TBAR T$(3) = = TBAR U$(1) = = U(NODE(I,1 ) ) U$(2) = = U(NODE(I,2)) U$(3) = = U(NODE(I,3)) V$( 1 ) = = V(NODE(I,1)) V$(2) = = V(NODE( I , 2) ) V$(3) = = V(NODE(I,3)) c A( 1 )=RC$(2)*ZC$(3)-RC$(3)*ZC$(2) A(2)=RC$(3)*ZC$(1)-RC$(1)*ZC$(3) A( 3)=RC$(1)*ZC$(2)-RC$(2 )*ZC$( 1 ) C C B(1) = = ZC$(2) • - ZC$(3) B(2) = = ZC$(3) • - ZC$( 1 ) B(3) = = ZC$( 1 ) • - ZC$(2) C( 1 ) = = RC$(3) • - RC$(2) C(2) = = RC$( 1 ) -- RC$(3) C(3) = = RC$(2) -- RC$( 1 ) C DEL2 = ABS(RC$( 1 ) *(ZC$(2)-ZC$(3)) + RC$(2)*(ZC$(3)-ZC$(1)) + RC$(3)*(ZC$ 1(1)-ZC$(2))) C RBAR = (RC$( 1 ) + RC$(2) + RC$(3))/3. ZBAR=(ZC$(1 ) + ZC$(2) + ZC$(3))/3. C C EPR=(B( 1 )*U$( 1 )+B(2)*U$(2)+B(3)*U$(3))/DEL2 C EP2=(C( 1 )*V$( 1 )+C(2)*V$(2)+C(3)*V$(3))/DEL2 C GRZ = (C(1 )*U$( 1 )+C(2)*U$(2)+C(3)*U$(3) + 1B( 1 )*V$( 1 )+B(2)*V$(2)+B(3)*V$(3))/DEL2 C C CALCULATE STRESSES AT EACH NODE IN ELEMENT I C DO 900 I$=1 ,3 C IF(RC$(I$).E0.O.O)G0 TO 895 C SIGR$(!$) = ( 1 .-XNU)*EPR+XNU*EPZ+XNU*U$(I$)/RC$( I$)-T$(1$) c SIGTH$(1$)=XNU*(EPR + EPZ) + ( 1 .-XNU)*U$(I $)/RC$(1$)-T$(1$) C SIGZ$(I$)=XNU*EPR+(1.-XNU)*EPZ+XNU*U$(I$)/RC$(I$)-T$(I$) C TAURZ$( 1$ ) = ( 1 . -2.*XNU)*GRZ/2. C GO TO 899 C 895 SIGR$(I$)=EPR+XNU*EPZ-T$(1$) C SIGTH$(I$)=SIGR$(1$) C SIGZ$(I$) = 2.*XNU*EPR+(1 .-XNU)*EPZ-T$(I $ ) C TAURZ$(!$)=(1.-2.*XNU)*GRZ/2. C 899 SIGRP(N(1$)) = SIGRP(N(1$)) + SIGR$(I $)*DEL2 SIGTHP(N(1$))=SIGTHP(N(1$))+SIGTH$(I $ ) *DEL2 SIGZP(N(1$))=SIGZP(N(1$))+SIGZ$(I$)*DEL2 TAURZP(N(1$))=TAURZP(N(I$) )+TAURZ$(I$)*DEL2 DEL2P(N(1$))=DEL2P(N(1$) )+DEL2 C C CYCLE FOR NEXT NODE IN ELEMENT I C 900 CONTINUE C C C C CYCLE FOR NEXT ELEMENT C 700 CONTINUE C C CALCULATION OF AVERAGE STRESSES AT ALL NODES C WRITE(6,2550) WRITE(6,2555) C C DO 960 I=1,NN C SIGRAV=SIGRP(I)/DEL2P( I ) SIGTHA = SIGTHP(I)/DEL2P( I ) SIGZAV = SIGZP(I)/DEL2P( I) TAURZA=TAURZP(I)/DEL2P(I) C C CALCULATE PRINCIPAL STRESSES AND VON MISES STRESSES C DO 950 I$=1 .6 STR(I$)=0.0 950 CONTINUE C STR(1)=SIGRAV STR(3)=SIGTHA STR(4 )=TAURZA STR(6)=SIGZAV C N$ = 3 NVEC=0 C CALL SYMAL(STR,N$,E$,I ERROR,NVEC) C VMS(I)=SQRT(((E$(3)-E$(2))**2+(E$(3)-E$(1))**2+ 1(E$(2)-E$( 1))**2)/2. ) C VMS(I)=VMS(I)*2.91*DTA WRITE(6,2560)1,SIGRAV.SIGTHA,SIGZAV,TAURZA,VMS(I ) C 960 CONTINUE C C PLOT CONTOURS OF VONMS AND TEMP NC=10 NX = 50 NY = 50 IDIMX=NX XMIN=0.0 YMIN=0.0 XMAX=10. v YMAX =10. DX=(XMAX-XMIN)/(NX-1) DY=(YMAX-YMIN)/(NY-1) C DO 970 1=1,NN RC(I) = RC(I )*5 .0 ZC(I)=ZC(I)*5.0 970 CONTINUE C C VMSMIN=VMS(1) C DO 975 1=1,NN 975 IF(VMSMIN.GT.VMS(I)) VMSMIN=VMS(I) C VMSMAX=VMS(1) C DO 978 1=1,NN 978 IF ( VMSMAX.LT.VMS(I) ) VMSMAX = VMS(I ) C DVMS=( VMSMAX-VMSMIN) C NRNG = 2 S=10. OPT=0.0 C CALL CGRID1(VONMS,IDIMX,NX,NY,XMIN,YMIN.DX.DY, 1 RC . ZC , VMS . NN, S , NRNG , OPT ) C C XSIZE=10. YSIZE=10. IOP(1)=1 I0P(2)=1 I0P(3)=0 I0P(4)=1 I0P(5)=0 I0P(6) = 1 I0P(7)=0 I0P(8) = 1 V0P(8)=.08 IDIMX=NX C M= 1 XNC=NC C DO 985 I=1,NC XI = I X1M=(XI-1 . )/(XNC-1 . ) CVAL(I)=VMSMIN+DVMS*XIM**M 985 CONTINUE CALL C0NTUR(XSIZE,YSIZE,VONMS,IDIMX,NX,NY, 1CVAL,NC,IOP,VOP) C C CALL PLOTND 990 WRITE(6,2001) C C C READEING FORMATS C 1005 F0RMAT(4(2X,13)) 1015 F0RMAT(2X,F5.2,2X,F7.5,2(2X,F7.3)) 1010 FORMAT(2X,F5.3,2(2X,E12.5),2X,F10.2) 1020 F0RMAT(2X.I4,2(2X,F1O,5)) 1025 F0RMAT(4(2X,14)) 1030 FORMAT(2X,I 3 ) 1040 FORMAT*10X,I3.6X.F15.4) C C WRITING FORMATS C 2001 FORMAT('SOLUTION FAILED') 2005 FORMAT( 'DETERMINANT IS',G16.7, '* 10**',I 20) 2010 F0RMAT(2X.'A FLEM SOLUTION FOR THERMAL DISPLACEMENTS',//,5X.'AND S 1TRESSES IN LEC-GAAS CRYSTALS',//) 2020 F0RMAT(5X. 'RADIUS (CM) = ' ,F7 . 5,/, 5X, 'VELOCITY (CM/SEC) ='.F7.5,/, 15X,'LENGTH (CM) =',F7.3,//) 2030 FORMAT(5X,'NU =',F5.3,/) 2035 F0RMAT(5X,'NUMBER OF NODES = ' ,I 3,/, 5X , 'NUMBER OF ELEMENTS = '.13 1,/,5X, 'NO.OF NODES WITH NO RADIAL DISPLACEMENT = ',13,/,5X, 'NO.OF 2N0DES WITH NO AXIAL DISPLACEMNT = ',13,/) 2040 FORMAT(10X,'NODES VS NODAL COORDINATES AND VS TEMPERATURE',//,6X,' 11 ' . 13X, 'R( I ) ' , 9X , 'Z( I ) ' , 4X , 'TEMEPERATURE ' , //) 2050 FORMAT(5X,I3,4X,2(3X,F10.5),2X,F15.4) 2060 FORMAT(' ',//, 15X,'SYSTEM TOPOLOGY',//,'ELEMENT NUMBER',11X,'NODE 1NUMBERS'.//) 2070 F0RMAT(5X.13,10X,3(5X,13)) 2080 FORMAT(' './/,5X,'NODES ON THE AXIS WITH NO RADIAL DISPLACEMENT',/ 1) 2085 FORMAT( ' ',/,5X,'NODES WITH NO AXIAL DISPLACEMENT',/) 2090 FORMAT(20X,13) 2099 FORMAT(/,5X,'ITERATION NUMBER = ',13,//) 2100 FORMAT( ' ' ,//, 10X, 'DISPLACEMENT FIELD: RADIAL AND AX I AL' ,/, 5X , ' NOD 1E NUMBER' ,4X, ' RADIAL'. 11X, 'AX IAL' ,/ ) 2110 F0RMAT(9X,I 3,2(5X,E15.6)) 2115 F0RMAT(5X,'CALCULATED DISPLACEMENT FIELD',/) 2120 F0RMAT(/,5X. 'NODES AND DISPLACEMENTS AT THE FREE BOUNDARY',/) 2500 FORMAT(' ',//,2X,'STRESSES CALCULATED AT ALL NODES IN EACH ELEMENT 1',//,2X,'ELEMENT',1X,'NODE',4X,'SIGMAR',4X,'SIGMATH',4X,'SIGMAZZ', 25X,'TAURZ',//) 2510 FORMAT(5X,13) 2520 FORMAT(1OX,I3,5X,4(E14.5)) 2550 FORMAT( ' ',/, 5X,'AVERAGE STRESSES AT EACH NODE',/) 2555 FORMAT(2X, 'NODE' ,2X, 'NEPN' ,7X. 'SIGRAV ' ,7X, 'SIGTHAV' ,6X, 'SIGZAV' ,8X 1.'TAURZAV',/) 2560 FORMAT(1X,I 3,3X,5(E14.5)) C C STOP END 412 XI.4 Finite Quadratic Element Program for the Stress Calculations c c C THIS PROGRAM CALCULATES THE DISPLACEMENTS AND STRESSES C IN LEC_GAAS CRYSTALS DURING GROWTH USING A FOEM METHOD C C C ELEMENTS ARE CALCULATED EXACTLY AND USING C C A OUINTIC NUMERICAL INTEGRATION FORMULA C C REAL*4 STR,E$,VMS DIMENSION STR(6),E$(3) INTEGERM I ERROR , NVEC . NEO$ REAL*4 E DIMENSION AB(78),E(12) INTEGER*4 NN,NEO INTEGER*4 NSOL.IPERM INTEGERM LLB , LUB , NHRS , I P , dEXP , ITER REAL*4 A,B,EPS,RES DIMENSION AA( 1 1 160),BB(90),IP(90),RES(90) DIMENSION Id(90,90) REAL*4 SK,SS,DES,DESDD,DET,SSN,DDES,DELDES DIMENSION SK(90,90),DES(90),SS(90),IPERM(180)„DESDD(90,90) DIMENSION SK$(12,12),SS$(12),T$(6),N(6),RC$(6),ZC$(6) DIMENSION A(3),B(3),C(3) DIMENSION RC(300),ZC(300).NODE(600,6),T(300),NSRD(100),NSAD(100) DIMENSION U(300),V(300).NSAS(300) DIMENSION U$(6),V$(6) DIMENSION SIGRP(300),SIGTHP(300),SIGZP(300),TAURZP(300),NEPN(300) DIMENSION DEL2P(300) DIMENSION NSSOOO) C DIMENSION W(7),XL(3,7) REALM ALPHA 1,ALPHA2,BETA 1,BETA2,W,XL C C C C READ NUMBER OF NODES AND NODES C C WITH SPECIFIED DISPLACEMENTS C C READ(5,1005)NN,NE.NNSRD,NNSAD C C INITIALIZE VARIABLES C NEO = 2*NN C DO 10 1 = 1, NEO SS(I) = o.o DES( I ) = 0.0 C DO 10 J=1,NEQ SK(I , J) = 0.0 10 CONTINUE DO 20 1=1,NN U( I ) = 0.0 V(I) = 0.0 RC(I) = 0.0 ZC(I) = 0.0 T(I) = 0.0 SIGRP(I) = 0.0 SIGTHP(I ) = 0.0 SIGZP(I) = 0.0 TAURZP(I ) =0.0 DEL2P(I)=0.0 20 CONTINUE C C READ CONSTANT AND COEFFICIENTS C READ(5,1010) XNU READ(5,1015) RO.VEL.XLE C DO 100 0 = 1.NN READ(5,1020) I,RC(I ) ,ZC( I ) WRITE(G.1020) I,RC(I ) ,ZC(I ) 100 CONTINUE C C READ SYSTEM TOPOLOGY C DO 105 I = 1.NE READ(5,1025)J,(NODE(J,1$),I$=1,6) WRITEI6,1025)J,(NODE(U,1$),I$=1,6) 105 CONTINUE C C C READ NODE NUMBERS WITH SPECIFIED DISPLACEMENT C DO 110 1 = 1 .NNSRD READ(5,1030) NSRD(I) WRITE(6.1030) NSRD(I) 110 CONTINUE C C C READ TEMPERATURES C DO 120 J = 1.NN READ(5, 1040) I ,T( I ) 120 CONTINUE C C PRINT DATA AND HEADLINES C WRITE(6.2010) WRITE(6.2020)RO.VEL.XLE WRITE(6,2030)XNU WRITE(6,2035)NN,NE,NNSRD C C C PRINT HEADLINES FOR DISPLACEMENTS C WRITE(G.2100) C C FORM MATRIX FOR INITIAL STRAIN AND STIFFNESS MATRIX C C GIVE VALUES TO WEIGHT FUNCTIONS AND C C NATURAL COORDINATES FOR NUMERICAL C C INTEGRARTION OF OUINTIC ORDER C ALPHA 1=.0597 1587 178977 BETA 1 =.470142064 105 1 15 ALPHA2=.797426985353087 BETA2 =.101286507323456 C W(1)=.225 W(2)=. 132 394152788506 W(3)=W(2) W(4)=W(2) W(5)=. 125939180544827 W(6)=W(5) W(7)=W(5) 414 XL( 1 , 1 )=1 ./3. XL( 1 ,2)=ALPHA1 XL( 1 ,3)=BETA1 XL( 1 , 4)=BETA1 XL( 1 ,5)=ALPHA2 XL(1,G ) =BETA2 XL( 1 ,7)=BETA2 C XL(2. 1 ) = 1,/3. XL(2.2)=BETA1 XL(2,3)=ALPHA1 XL(2,4)=BETA1 XL(2,5)=BETA2 XL(2,6)=ALPHA2 XL(2,7)=BETA2 C DO 140 1=1,7 XL(3,I)=1.-(XL(1,I)+XL(2.I)) 140 CONTINUE C C DO GOO K=1,NE C c DO 180 I$=1 ,6 N(I$)=NODE(K,1$) RC$(I$)=RC(NODE(K,1$ ) ) ZC$(I$)=ZC(NODE(K,1$) ) 180 CONTINUE C C DO 185 I$=1,3 T$(I$)=T(NODE(K,1$)) I$R=I$+3 I$1=I$+1 IF(I$1.GT.3) 1$ 1 = 1$ 1-3 T$(I$R)=(T(NODE(K,1$))+T(NODE(K,I$1)))/2. C 185 CONTINUE C GO TO 189 TBAR=0.0 DO 187 I$=1,G TBAR = TBAR + T(NODE(K.I $ ) ) 187 CONTINUE DO 188 I$=1,G T$(1$ )=TBAR/6. 188 CONTINUE 189 DO 190 I$=1,3 J$=I$+1 K$=J$+1 IF(U$.GT.3) d$=J$-3 IF(K$.GT.3) K$=K$-3 C A(I$)=RC$(d$)*ZC$(K$)-RC$(K$)*ZC$(d$) B(I$)=ZC$(d$)-ZC$(K$) C(I$)=RC$(K$)-RC$(U$) C 190 CONTINUE C SR = RC$(1) + RC$(2) + RC$(3) RBAR = SR/3. ZBAR = (ZC$(1) + ZC$(2) + ZC$(3))/3. DE L 2 = ABS(RC$( 1 )*(ZC$(2) - ZC$(3)) + RC$(2)*(ZC$(3 ) - ZC$(1)) + 1RC$(3)*(ZC$( 1) - ZC$(2))) C C C FORM MATRIX FOR INITIAL STRAIN IN ELEMENT K C C FORM MATRIX FOR INITIAL STRAIN C IN ELEMENT K NODES 1 TO 3 C 415 DO 230 I$= 1 .3 C K$ = 2*I$-1 SS$(K$)=T$(I$)*B(I$)*RC$(1$)/6. C K$=2*I$ SS$(K$)=T$(I$)*C(I$)*RC$(I$)/6. C C FORM STIFFNESS MATRIX FOR ELEMENT K C NODES 1 TO 3 C DO 230 J$=1,3 XNT=0.0 C C C INTEGRATE NUMERICALLY TERM IN LI'S C . DO 195 1=1,7 C XNT = XNT+W(I)*XL(I$,I)*(2.*XL(I$,I)-1.)* 1XL(d$,I)*(2.*XL(J$,I)-1.)/ 2(RC$(1)*XL(1.I ) + RC$(2)*XL(2.I) + RC$(3)* XL(3.I)) C 195 CONTINUE C IF(I$.EQ.d$)GO TO 210 C X1=-(SR+RC$(I$)+RC$(d$))/(30.*DEL2) X2 = - . 5 GO TO 215 C 210 X1 = (SR + 2 . *RC$ (1$))/(10.*DEL2) X2= 1 . 215 K$=2*I$-1 L$=2*d$-1 C SK1=((1.-XNU)*B(I$)*B(d$)+(.5-XNU)*C(I$)*C(d$))*X1 C SK$(K$,L$)=SK1+X2*(B(I $ )+B(d$))*XNU/15. + (1 .-XNU)* 1DEL2*XNT/2. C K$=2*I$ C SK$(K$,L$) = (XNU*C(I$)*B(d$) + ( .5-XNU)*C(d$)*B(I $))* 1X1+X2*XNU*C(I $)/15. C L$=2*d$ C SK$(K$,L$)=((1.-XNU)*C(I$)*C(d$)+(.5-XNU)*B(I$)*B(d$))*X1 C K$=2*I$-1 C SK$(K$.L$)=(XNU*B(I$)*C(d$)+(.5-XNU)*C(I$)*B(d$))* 1X1+X2*XNU*C(d$)/15. C 230 CONTINUE C C FORM INITIAL STRAIN MATRIX FOR C NODES 4 TO 6 C DO 250 I$= 1 ,3 C 1$ 1 = I$+ 1 IF(I$1.GT.3) I$1=I$1-3 C I$R=I$+3 C K$=2*I$R-1 C SS$(K$ )=T$(I$R)*(B(I$)*(SR+RC$(I$1 ) ) + 1B(I$1)*(SR+RC$(1$))+DEL2)/6. C 4 16 K$=2*I$R C SS$(K$)=T$(I$R)*(C(I$)*(SR+RC$(I$1))+ 1C(I$1)*(SR+RC$(I$)))/6. C C FORM MATRIX FOR NODES 4 TO 6 C DO 250 J$= 1 , 3 C J$1=J$+1 IF(J$ 1 .GT.3) J$1=J$1-3 C C C INTEGRATION OF TERM IN LI'S AND LJ'S C YNT=0.0 C DO 235 1=1,7 C YNT=YNT+W(I)*XL(I$,I)*XL(I$1,I)*XL(U$,I)*XL(J$1,I)/ 1(RC$(1)*XL(1,I) + RC$(2)*XL(2,I) + RC$(3 ) *XL(3,I ) ) C 235 CONTINUE C SR1=SR + RC$(I$1 ) + RC$(d$1 ) SR2=SR+RC$(I$)+RC$(d$) SR3=SR+RC$(I$1 ) + RC$(J$ ) SR4 = SR + RC$(I$) + RC$(d$1 ) C Y11=B(I$)*B(J$)*SR1+B(I$1)*B(J$1 )*SR2 + 1B(1$)*B(J$1)*SR3+B(I$1)*B(J$)*SR4 C Y44=C(I$)*C(d$)*SR1+C(I$1)*C(d$1 )*SR2 + 1C( 1$ )*C(d$ 1 )*SR3+C( 1$ 1 ) *C( J$ )*SR4 C Y14 = B(I$)*C(d$)*SR1+B(1$1)*C(d$1 )*SR2 + 1B(I$)*C(J$1 )*SR3+B(I$1)*C(J$)*SR4 C Y4 1=C(I$)*B(d$)*SR1+C(I$1)*B(d$1)*SR2+ 1C(I$)*B(d$1 )*SR3+C(I$1)*B(J$)*SR4 C IF(I$.EO.d$) GO TO 240 C IF(I$1.EO.J$) GO TO 238 C C OTHERWISE d$+1=I$, NEXT C Y11=Y1 1+B(I $ 1 )*B(J$)*SR4 C Y44=Y44+C(I$1)*C(d$)*SR4 C Y14=Y14+B(I$1)*C(d$)*SR4 C Y41=Y4 1+C(I$1)*B(J$)*SR4 C Yl11=B(I$) + 2.*B(I$1) + 2.*B(J$)+B(J$1 ) Yl12=2.*C(J$)+C(d$1) YI21=C(1$ ) + 2.*C(1$ 1 ) C GO TO 245 C 238 Y11=Y11+B(I$)*B(d$1)*SR3 C Y44=Y44+C(I$)*C(d$1)*SR3 C Y14=Y14+B(I$)*C(J$1)*SR3 C Y41=Y41+C(I$)*B(J$1)*SR3 C YI 1 1 = 2 . *B(I$) + B(I$1)+B(J$) + 2.*B(J$1) YI 12=C(J$)+2.*C(d$1) YI2 1=2 . *C(I$)+C(I$1) C GO TO 245 C 240 Y 1 1=Y 1 1+B(I$)*B(J$)*SR1+B(I$1)*B(d$1)*SR2 C Y44=Y44+C(I$)*C(J$)*SR1+C(I$1)*C(d$1)*SR2 C Y14 = Y14 + B(I$)*C(J$)*SR1+B(I$1)*C(d$1)*SR2 C Y4 1=Y4 1+C(I$)*B(J$)*SR1+C(I$1)*B(J$1)*SR2 C YI 1 1=2 . *(B(I$)+B(I$1)+B(J$)+B(J$1)) YI 12 = 2 . *(C(U$)+C(d$1)) YI21=2 . *(C(I$)+C(I$1)) C C 245 J$R=J$+3 C K$=2*I$R-1 L$=2*d$R-1 C SK$(K$,L$)=(((1.-XNU)*Y11+(.5-XNU)*Y44)/ 1DEL2+XNU*YI11)/7.5+8.*(1.-XNU)*DE L2 * YNT C K$=2*I$R C SK$(K$,L$ ) = ((XNU*Y41 + ( .5-XNU)*Y14)/DEL2 + 1XNU*YI21)/7.5 C L$=2*d$R C SK$(K$,L$)=((1.-XNU)*Y44+(.5-XNU)*Y11)/(DEL2*7.5) C K$ = 2*I$R- 1 C SK$(K$,L$)=((XNU*Y14+(.5-XNU)*Y41)/DEL2+ 1XNU*YI12)/7.5 C 250 CONTINUE C C STIFFNESS MATRIX FOR MIXED NODES C DO 290 I$= 1 ,3 DO 290 J$= 1 .3 C d$R=d$+3 C U$1=J$+1 IF(J$1.GT.3) d$1=U$1-3 C C C INTEGRATE TERM IN LU'S AND LQ'S C ZNT=0.0 C DO 255 1=1,7 C ZNT=ZNT+W(I)*XL(1$,I)*(2.*XL(I$,I)-1.)* 1XL(d$,I)*XL(d$1.I)/ 2(RC$( 1 ) *XL( 1 , I ) + RC$(2)*XL(2,I) + RC$(3)*XL(3.I)) C 255 CONTINUE C IF(I $.EQ.0$ ) GO TO 270 C IF(1$.EO.d$1) GO TO 260 C C OTHERWISE 0 NOT = I AND 1+1 NOT =Q,NEXT 4 18 21 1 = (B(d$)+B(J$1 ))*(4.*RC$(I$)-SR)-1(RC$(J$)*B(J$1 ) + RC$(J$1 )*B(J$) ) C Z44 = (C(J$ )+C(J$1 ))*(4.*RC$(I$)-SR)-1(RC$(J$)*C(J$1 ) + RC$(J$1 )*C(J$) ) C ZI 1 1 = -(B(J$) + B(d$1)+B(1$)) ZI12 = -(C(J$ )+C(J$1)) ZI21 = -C(1$ ) C GO TO 280 C 2G0 Z11 = (3.*B(J$)-B(J$1 ) )*SR+( 1 1 . *B(J$) + 14.*B(J$1 ) )*RC$(I$)-B(J$1)*RC$(J$) C Z44=(3.*C(J$)-C(J$1 ) )*SR+(1 1 .*C(J$) + 14.*C(J$ 1 ) )*RC$(I$)-C(J$1)*RC$(J$) C ZI 1 1=2.*B(I $ ) + 2.*B(J$) ZI12=2.*C(U$)-C(J$1) ZI2 1 = 3.*C(1$) C GO TO 280 C 270 Z11 = (3.*B(d$1 )-B(J$) )*SR+(4.*B(J$) + 111 .*B(d$1 ) )*RC$(I$)-B(d$)*RC$(J$1 ) C Z44 = (3.*C(d$1 )-C(d$ ) )*SR+(4.*C(d$) + 111 .*C(d$1))*RC$(I$)-C(J$)*RC$(d$1 ) C ZI11 = 2.*B(I$) + 2.*B(J$1 ) ZI12 = -C(d$) + 2.*C(d$1 ) ZI21=3.*C(I$) 280 K$=2*I$-1 L$ = 2*vJ$R-1 C SK$(K$,L$ ) = (((1.-XNU)*B(I$)*Z11 + ( .5-XNU )*C(I$)*Z44)/ 1DEL2+XNU*ZI11)/30.+2.*(1.-XNU)*DEL2*ZNT C SK$(L$.K$)=SK$(K$,L$) C K$=2*I$ C SK$(K$.L$) = ((XNU*C(I$)*Z11 + ( .5-XNU)*B(1$ )*Z44)/ 1DEL2+XNU*ZI21)/30. C C SK$( L$ ,K$ ) =SK$(K$ , L$ ) C L$=2*J$R C SK$(K$,L$ ) = (( 1 .-XNU)*C(I$)*Z44 + ( .5-XNU ) *B(I $ ) *Z 1 1 )/ 1(30.*DEL2) C SK$(L$,K$) = SK$V(K$,L$) C K$=2*I$-1 C SK$(K$,L$) = ((XNU*B(I$)*Z44 + ( .5-XNU )*C(I$)*Z11)/ 1DEL2+XNU*ZI12)/30. C C SK$(L$,K$)=SK$(K$,L$) C C 290 CONTINUE C GO TO 355 WRITE(6,2510)K DO 300 I$=1 , 12 WRITE(6.301) (SK$(1$,d$) ,J$ = 1 . 12) e DO 300 d$ = 1 . 1$ I$d$=I$*(I$-1)/2+d$ AB(I$d$)=SK$(I$,d$) 300 CONTINUE 301 FORMAT(12F8.5) C NVEC=0 NEQ$= 12 CALL SYMAL ( AB,NE0$,E,I ERROR,NVEC ) DO 350 I$= 1 , 12 WRITE(6,351) I$,E(I$) 350 CONTINUE C 351 F0RMAT(I3,F12.6) WRITE(6,25G0) I ERROR C ASSEMBLE MATRICES WITHOUT REGARDING B.C. C 355 DO 500 I$=1 ,6 C I = N(I$) C M$ = 2*1$ - 1 M = 2*1 - 1 SS(M) = SS(M) + SS$(M$) C M$ = 2*1$ M = 2*1 SS(M) = SS(M) + SS$(M$) C C DO 500 d$ = 1,6 C J = N(J$) C M$ = 2*1$ - 1 M = 2*1 - 1 N$ = 2*d$ - 1 N1 = 2*d - 1 SK(M.N1) = SK(M.N1) + SK$(M$,N$) C N$ = 2*d$ N1 = 2*J SK(M,N1) = SK(M,N1) + SK$(M$.N$) C M$ = 2*1$ M = 2*1 SK(M.N1) = SK(M,N1) + SK$(M$,N$) C N$ = 2*d$ - 1 N1 = 2*J - 1 SK(M,N1) = SK(M,N1) + SK$(M$,N$) C C ASSEMPLE ELEMENTS OF THE NEXT NODE IN ELEMENT K C 500 CONTINUE C C CYCLE FOR NEXT ELEMENT C 600 CONTINUE C C MODIFY SK AND SS TO ACCOUNT FOR NODES WITH SPECIFIED DIStLACEMENT C DO 700 1=1,NNSRD d = 2*NSRD(I) - 1 DO 690 L=1 ,NEO SK(d,L)=0.0 SK(L,d)=0.0 690 CONTINUE SS(d) = 0.0 SK(d.d)=1. 700 CONTINUE C c DO 706 1=1.1 L = 2*I DO 705 J=1.NEQ SK(J,L)=0.0 SK(L,J)=0.0 705 CONTINUE SS(L)=0.0 SK(L . L ) = 1 . 706 CONTINUE C GO TO 708 NS0L=1 CALL FSLE(NEO.NEO,SK,NSOL,NEO,SS.DES.I PERM,NEO,DESDD,DET,JEXP) WRITE (6,2005) DET,JEXP GO TO 720 C 708 LUB=41 LLB=41 LK=2*LLB+LUB LIN=NEQ*(LK+1) DO 710 1=1,NEO BB(I ) = SS(I ) DO 710 J=1,NEO I J(I,J)=LK*J+I-LLB IF (Id(I,J) GT.LIN) GO TO 710 A A ( I J ( I , J ) ) = SK ( I , J ) 710 CONTINUE C C CALL FOR GBAND C NHRS=+1 ITER=0 CALL GBAND(AA.BB,NEO,LLB,LUB.NHRS,IP,DET,JEXP, BB,RES,ITER,EPS) WRITE(6,2005) DET,JEXP C C PRINT DISPLACEMENTS C C 720 WRITE(6,2115) DO 790 1=1,NN J = 2*1 - 1 U(I ) = BB(J) K = 2*1 V(I) = BB(K) C WRITE(6,2110)1,U(I),V(I) 790 CONTINUE C C C CALCULATE STRESSES AT ALL NODES IN EACH ELEMENT C C DO 950 I = 1,NE C DO 810 I$= 1 .6 N(I$)=NODE(I,1$) RC$(I$)=RC(NODE(I,1$)) ZC$(I$)=2C(N0DE(I,1$)) U$(I $)=U(NODE(I,1$)) V$(I$)=V(NODE(I,1$)) 810 CONTINUE C DO 815 I$=1 ,3 T$(I $)=T(NODE(I,1$ ) ) C I$R=I$+3 I$1=I$+1 IF(I$1.GT.3) 1$1=1$1-3 T$(I$R)=(T(NODE(I,1$))+T(NODE(I.I$1)))/2. C 815 CONTINUE C GO TO 819 421 TBAR=0.0 DO 817 I$=1 ,6 TBAR=TBAR+T(NODE(I,1$) ) 817 CONTINUE DO 818 I$=1 .6 T$(I$)=TBAR/6. 818 CONTINUE C 819 DO 820 I$=1 ,3 J$=I$+ 1 K$=J$+ 1 C IF(J$.GT.3) J$=J$-3 IF(K$.GT.3) K$=K$-3 C A(I$)=RC$(d$)*ZC$(K$)-RC$(K$)*ZC$(d$) B(I$)=ZC$(d$)-ZC$(K$) C(I$)=RC$(K$)-RC$(d$) C 820 CONTINUE C DEL2=ABS(RC$(1)*(ZC$(2)-ZC$(3)) + RC$(2)*(ZC$(3)-ZC$(1))+RC$(3)*(ZC$ 1( 1)-ZC$(2))) C RBAR=(RC$(1)+RC$(2)+RC$(3))/3. ZBAR=(ZC$(1)+ZC$(2)+ZC$(3))/3. C C C CALCULATION OF STRESSES AT C C NODES 1 TO 3 IN ELEMENT C DO 850 IS= 1 .3 C I$M1=I$-1 IF(I$M1.EO.O) I$M1=3 C I$1=I$+1 IF(I$1.GT.3) 1$ 1 = 1$1-3 C I$2=I$+2 IF(I$2.GT.3) I$2=I$2-3 C I$R=I$+3 I$RM1=I$M1+3 C C EPR=(3.*B(I$)*U$(I$)-B(I$1)*U$(I$1)-B(I$2)*U$( I $2) 1+4.*(B(I$1 )*U$(I$R)+B(I$M1)*U$(I$RM1)))/DEL2 C EPZ=(3.*C(I$)*V$(I$)-C(I$1)*V$(I$1)-C(I$2)*V$(I$2) 1+4.*(C(I$1)*V$(I$R)+C(I$M1)*V$(I$RM1)))/DEL2 C GRZ1=C(I$)*U$(I$)+B(I$)*V$(1$) GRZ2=C(I$1)*U$(I$R)+B(I$1)*V$(I$R) GRZ3=C(I$M1)*U$(I$RM1)+B(I$M1)*V$(I$RM1) GRZ4=C(1$1)*U$(1$1)+C(I$2)*U$(I$2) GRZ5=B(I$1)*V$(I$1)+B(I$2)*V$(I$2) C GRZ=(3.*GRZ1+4.*(GRZ2+GRZ3)-GRZ4-GRZ5)/DEL2 C c IF(RC$(I$).EQ.O.O) GO TO 830 C EPT=U$(I$)/RC$(1$) GO TO 840 C 830 EPT=EPR C 840 SIGR$ = ( 1 .-XNU)*EPR+XNU*(EPT + EPZ)-T$(I$) C 422 SIGTH$=XNU*(EPR+EPZ)+(1.-XNU)*EPT-T$(I $ ) C SIGZ$=XNU*(EPR+EPT ) + (1.-XNU)»EPZ-T$( 1$ ) C TAURZ$=(.5-XNU)*GRZ C C C C 869 SIGRP(N(1$ ) )=SIGRP(N(1$))+SIGR$*DEL2 SIGTHP(N(1$))=SIGTHP(N(I$))+SIGTH$*DEL2 SIGZP(N(1$))=SIGZP(N(1$))+SIGZ$*DEL2 TAURZP(N(1$))=TAURZP(N(1$))+TAURZ$*DEL2 DEL2P(N(1$))=DEL2P(N(1$))+DEL2 C C CYCLE FOR NEXT NODE IN ELEMENT I C 850 CONTINUE C C CALCULATION OF STRESSES AT C C NODES 4 TO 6 IN ELEMENT C DO 900 I$= 1 .3 C I$R=I$+3 C I$1=I$+1 IF(I$1.GT.3) I$1=I$1-3 C I$2=I$+2 IF(I$2.GT3) I$2=I$2-3 C I$R1=I$1+3 C I$M1=I$-1 IF(I$M1.EO.O) I$M1=3 C I$RM1=I$M1+3 C EPR=(B(I$)*U$(I$)+B(I$1)*U$(I$1)-B(I$2)*U$( I$2) + 12.*(B(I$)+B(I$1))*U$(I$R))/DEL2+ 22 . *B(I$M1 )*(U$(I$R1)+U$(I$RM1))/DEL2 C EPZ=(C(I$)*V$(I$)+C(I$1)*V$(I$1)-C(I$2)*V$(I$2)+ 12.*(C(I$) + C(I$1 ))*V$(I$R))/DEL2+ 22.*C(I$M1)*(V$(I$R1)+V$(I$RM1))/DEL2 C GRZ1=C(I$)*U$(I$)+C(I$1)*U$(I$1)-C(I$2)*U$(I$2) GRZ2=B(I$)*V$(I$)+B(I$1)*V$(1$1)-B(I$2)*V$(I$2) GRZ3=(C(I$)+C(I$1))*U$(I$R)+(B(I$)+B(I$1))*V$(I$R) GRZ4=B(I$M1)*(V$(I$R1)+V$(I$RM1)) GRZ5 = C(I$M1 )*(U$(I$R1)+U$(I$RM1)) C GRZ=(GRZ1+GRZ2+2.*(GRZ3+GRZ4+GRZ5))/DEL2 C C IF(RC$(I$R) .EO.0.0) GO TO 880 EPT=U$(I$R)/RC$(I$R) C GO TO 890 C 880 EPT=EPR C 890 SIGR$ = ( 1 .-XNU)*EPR + XNU*(EPT + EPZ)-T$( I$R) C SIGTH$ = XNU*(EPR+EPZ) + (1.-XNU)*EPT-T$ ( I $R) C SIGZ$=XNU*(EPR+EPT)+(1.-XNU)*EPZ-T$(I$R) C TAURZ$=( ,5-XNU)*GRZ C 423 c 899 SIGRP(N(I$R))=SIGRP(N(I$R))+SIGR$*DEL2 SIGTHP(N(I$R))=SIGTHP(N(I$R))+SIGTH$*DEL2 5IGZP(N(I$R))=SIGZP(N(I$R))+SIGZ$*DEL2 TAURZP(N(I$R))=TAURZP(N(I$R))+TAURZ$*DEL2 DEL2P(N(I$R))=DEL2P(N(I$R))+DEL2 C 900 CONTINUE C C 950 CONTINUE C C CALCULATION OF AVERAGE STRESSES AT ALL NODES C WRITE(6.2550) WRITE(6,2555) C C DO 960 I=1,NN C SIGRAV=SIGRP(I)/DEL2P(I) SIGTHA = SIGTHP(I)/DEL2P(I ) SIGZAV=SIGZP(I)/DEL2P(I) TAURZA = TAURZP(I )/DEL2P(I ) C C CALCULATE PRINCIPAL AND VON MISES STRESSES C DO 955 I$=1 .6 STR(I$)=0.0 955 CONTINUE C STR(1)=SIGRAV STR(3)=SIGTHA STR(4 )=TAURZA STR(6)=SIGZAV C N$ = 3 NVEC=0 C CALL SYMAL(STR,N$,E$,I ERROR,NVEC) C VMS=SORT(((E$(3)-E$(1))**2+(E$(2)-E$(1))**2+(E$(3)-E$(2))**2)/2.) C WRITE(6,2560)I,SIGRAV,SIGTHA,SIGZAV,TAURZA,VMS C 960 CONTINUE C C C READEING FORMATS C 1005 F0RMAT(4(2X,13)) 1015 F0RMAT(2X.F5.2,2X,F7.5,2X,F7.3) 1010 FORMAT(2X,F5 3,2(2X,E12.5),2X,F10.2) 1020 F0RMAT(2X,13,2(2X,F10.5)) 1025 F0RMAT(7(2X,13)) 1030 FORMAT(2X,13) 1040 FORMAT*10X,I3.6X.F15.4) C C WRITING FORMATS C 2001 FORMAT*'SOLUTION FAILED') 2005 FORMAT*'DETERMINANT IS ' , G16 . 7 , '* 10**'.120) 2010 F0RMAT(2X,'A FLEM SOLUTION FOR THERMAL DlSPLACEMENTS',//,5X,'AND S 1TRESSES IN LEC-GAAS CRYSTALS',//) 2020 F0RMAT(5X,'RADIUS (CM) = ',F7.5,/,5X,'VELOCITY (CM/SEC) ='.F7.5,/, 15X,'LENGTH (CM) =',F7.3,//) 2030 FORMAT(5X,'NU =',F5.3,/) 2035 FORMAT(5X, 'NUMBER OF NODES = ' ,I 3,/,5X, 'NUMBER OF ELEMENTS = ',13 1,/.5X, 'NO.OF NODES WITH NO RADIAL DISPLACEMENT = ' ,I 3,/, 5X , 'NO.OF 2NODES WITH NO AXIAL DISPLACEMNT = ',13./) 2040 F0RMAT( 10X . 'NODES VS NODAL COORDINATES AND VS TEMPERATURE',//.6X, ' 1I'.13X,'R(I)'.9X,'Z(I)',4X,'TEMEPERATURE'.//) 2050 F ORMAT(5X,I 3,4X.2(3X,F10.5).2X,F15.4) 2060 FORMAT( ' ',//. 15X,'SYSTEM TOPOLOGY',//.'ELEMENT NUMBER', 11X.'NODE 1 NUMBERS ' ,//) 2070 F0RMAT(5X,13,10X,3(5X,13)) 2080 FORMAT(' './/,5X,'NODES ON THE AXIS WITH NO RADIAL DISPLACEMENT',/ 1 ) 2085 FORMAT(' ',/,5X,'NODES WITH NO AXIAL DISPLACEMENT',/) 2090 F0RMAT(20X,13) 2099 F0RMAT(/.5X,'ITERATION NUMBER = ',13,//) 2100 FORMAT( ' ' .//, 10X, 'DISPLACEMENT FIELD: RADIAL AND AX IAL ' ,/,5X, 'NOD 1E NUMBER' ,4X, 'RADIAL' , 1 1X, 'AXIAL',/) 2110 FORMA T(9X,I3,2(5X,E15.6)) 2115 F0RMAT(5X,'CALCULATED DISPLACEMENT FIELD',/) 2120 FORMAT(/,5X,'NODES AND DISPLACEMENTS AT THE FREE BOUNDARY',/) 2500 FORMAT(' ',//,2X,'STRESSES CALCULATED AT ALL NODES IN EACH ELEMENT 1',//,2X,'ELEMENT',1X,'NODE',4X,'SIGMAR',4X.'SIGMATH'.4X,'SIGMAZZ'. 25X,'TAURZ',//) 2510 FORMAT(5X,I 3) 2520 FORMAT(10X.13,5X,4(E14.5)) 2550 FORMAT( './.5X,'AVERAGE STRESSES AT EACH NODE',/) 2555 FORMAT(2X , 'NODE' ,7X, 'SIGRAV ,7X. 'SIGTHAV' ,6X. 1'SIGZAV,8X,'TAURZAV',/) 2560 FORMAT*1X,13,3X,5(E14.5)) C C STOP END 425 XI.5 Program for the Calculations of the MRSS-CRSS in (010) Sections c c C THIS PROGRAM CALCULATE AND PLOT C C C MAXIMUN RESOLVED SHEAR STRESSES C C DIMENSION T(600),RSSM$(5) DIMENSION SIGR(500),SIGZ(500),SIGT(500).TARZ(500) DIMENSION RC(500),ZC(500).RSSM(500),RSMG(50,50) DIMENSION CVAL(20),I0P(8).V0P(8) REAL TAUCR.T.DTA REAL SIGR,SIGT,SIGZ,TARZ,RC,ZC,RSSM,RSMG,CVAL,VOP REAL C.XMIN,YMIN,DX.DY,DRSM,XSIZE.YSIZE INTEGER NN,IDIMX,NX,NY,NRNG,NC,IOP,IMAX$ REAL TAUCRM,TAUCTE,TAUCRJ,RSSM$ REAL XIN.XI,XCN C DTA=200. C NN= 1 11 C WRITE(6 , 1 1 ) 11 F0RMAT(9X.'NODE',10X,'MAX.SHEAR STRESS(MPA)'./) DO 20 I=1,NN 20 READ(5,21) SIGR(I),SIGT(I),SIGZ(I).TARZ( I ) 21 F0RMAT(7X,4(E14.5)) C DO 30 J=1,NN READ(5,31) I , RC(I),ZC(I) RC(I ) = RC(I ) *5.0 ZC(I)=ZC(I)*5.0 30 CONTINUE 31 FORMAT(2X,I4,2(2X,F10.5)) C 39 DO 40 I=1,NN 40 READ(5.41 )d,T(<J) C 41 FORMAT(1OX,I3.GX.F15.4) C C DO 50 1=1.NN C TAUCR=TAUCRJ(DTA,T(I)) C RSSM(I)=0.0 SRT=(SIGR(I)-SIGT(I))*2.91*DTA/SQRT(6.) SZT=(SIGZ(I)-SIGT(I))*2.91*DTA/SQRT(6.) SRZ=(SIGR(I )-SIGZ(I))*2.91*DTA/SORT(G. ) TARZ(I )=TARZ(I)* 2.91 *DTA/SORT(6. ) C C RSSM$(1)=ABS(-SRT+TARZ(I)J-TAUCR C RSSM$(2)=ABS(-SRT-TARZ(I))-TAUCR C RSSM$(3)=ABS(SZT+TARZ(I))-TAUCP C RSSM$(4)=ABS(SZT-TARZ(I))-TAUCR C RSSM$(5)=ABS(SRZ)-TAUCR C 426 RSMAX$=RSSM$(1) IMAX$=1 C DO 45 I$=1,5 IF(RSMAXS.LT.RSSM$(I$)) GO TO 43 GO TO 45 43 RSMAX$=RSSM$(1$) IMAX$=I$ 45 CONTINUE C RS5M(I)=RSMAX$ C IF(RSSM(I).LT.0.0) RSSM(I)=0.0 C C WRITE(6,51) I,RSSM(I),IMAX$ C 50 CONTINUE C C 51 FORMAT(10X,I3,12X,F15.6,4X,I3) C C CALCULATE RSMMAX AND RSMMIN C RSMMAX=RSSM(1) DO 70 1=1,NN 70 IF(RSMMAX.LT.RSSM(I)) RSMMAX=RSSM(I) C RSMMIN=RSSM(1) DO 80 1=1.NN 80 IF(RSMMIN.GT.RSSM(I ) ) RSMMIN=RSSM(I) WRITE(6,82) RSMMAX 82 FORMAT(/,5X, 'RSMMAX=' ,F 15 . 7,/) C C CALCULATE REGULAR GRID C NX = 50 NY = 50 IDIMX=NX C C XMIN=0.0 YMIN=0.0 XMAX=10. YMAX=10. DX = (XMAX-XMIN)/(NX-1 ) DY = (YMAX-YMIN)/(NY-1 ) NRNG=2 C= 10. 0PT=O.O C CALL CGRID1(RSMG.IDIMX,NX.NY,XMIN,YMIN, 1 DX.DY,RC,ZC,RSSM,NN,C,NRNG,OPT) C C PLOT GRID C 88 XSIZE=10. YSIZE=10. NC= 10 IOP(1)=1 I0P(2)=1 I0P(3)=0 I0P(4 ) = 1 I0P(5)=O I0P(6)=1 I0P(7)=O I0P(8)=1 V0P(8)=.10 C DRSM=(RSMMAX-RSMMIN) M= 1 XNC=NC 427 c DO 90 1 = 1 ,NC XI = 1 XIN=(XI-1 . )/(XNC-1 . ) CVAL(I) = RSMMIN->-DRSM*XIN**M 90 CONTINUE C CALL CONTUR(XSIZE,YSIZE,RSMG,IDIMX,NX,NY, 1 CVAL,NC,IOP.VOP) C CALL PLOTND RETURN END C C FUNCTION TO CALCULATE C C CRITICAL STRESS AT T(I) C REAL FUNCTION TAUCRM(DTA,TT) REAL DTA,TT TAUCRM=6.8647E-2*EXP(5.1989E3/(1511-DTA*(1.-TT))-3.4407) RETURN END C C CRITICAL STRESSES FOR C C TE-DOPED GAAS (M AND B) C REAL FUNCTION TAUCTE(DTA,TT) REAL DTA,TT C TAUCTE=.39227*EXP(5.9205E3/(1511-DTA*(1 .-TT))-3.9183 ) C RETURN END C C CRITICAL STRESSES FOR YIELD C REAL FUNCTION TAUCRJ(DTA,TT) REAL DTA,TT C TAUCR<J=.5625*EXP(5.0596E3/(1511-DTA*(1 .-TT) )-3.3485) C RETURN END 428 XI.6 Program for the Calculations of MRSS-CRSS in (001) Sections c c C THIS PROGRAM PLOT RESOLVED SHEAR C C STRESSES IN POLAR COORDINATES C C DIMENSION T( 1 1 ) ,RSS$( 10) DIMENSION RSSG(50,50),RSS(700),XP(700),YP(700) DIMENSION V0P(8),I0P(8),CVAL(10) DIMENSION RS(11),TS(11),ZS(11),RZS(11),R0(11) REAL RS.TS.ZS,RZS,RO,RSSG,RSS,XP,YP REAL THETA,DTHETA,VOP,CVAL,C INTEGER IOP.NC,IDIMX,NN,NV.NX,NY.IMAX$ REAL TAUCR.TAUCTE,TAUCRM,TAUCRJ,RSS$ REAL XI.XNC.XIN C C C DTA=200. C WRITE(6.2) 2 F0RMAT(//,5X, 'RESOLVED SHEAR STRESS ' ,/'MAX IMUN COMPONENT',/) C READ(5,6) ZC 6 FORMAT(20X.F10.5) C WRITE(6,7) ZC 7 FORMAT(10X,'HEIGHT = '.F10.5,/) C WRITE(6,8) 8 FORMAT(//,5X,'NV',8X,'RSS(MPA)',6X,'MODE',/) C C C READ STRESSES FROM R0=0 TO 1. AT Z C DO 10 1=1,11 10 READ(5,11) RS(I ) ,TS(I ) ,ZS(I ) ,RZS(I ) 11 F0RMAT(7X,4(E14.5)) C DO 15 1=1,11 READ(5 , 16 ) T( I ) WRITE(6, 16) T(I ) 15 CONTINUE 16 FORMAT(19X.F15.4) C C C CALCULATE NUMBER OF POINTS OVER CIRCLE C C OF RADIUS RO WHERE STRESSES WILL BE C C CALCULATED C NV= 1 C DO 40 1=1,10 I$1=I+1 C TAUCR=TAUCRJ(DTA,T(I$1)) C RO(I)=5 . * I/ 10 NN=4*I DTHETA = 6.2831853/NN RSB=(RS(I$1 )-TS(I$1 ))*2.91*DTA/SQRT(6. ) ZSB=(ZS(I$1 )-TS(I$1 ) )*2.91*DTA/S0RT(6. ) RZS(1$ 1 )=RZS(I$1 )*2.91*DTA/SQRT(6. ) DD 40 d=1,NN THETA=DTHETA*(J-1) XP(NV)=R0(I)*C0S(THETA)+5. YP(NV)=R0(I)*SIN(THETA)+5. PI4= .7853982 TERM1=-RSB*C0S(2.*THETA) TERM2=SQRT(2.)*RZS(1$1)*C0S(THETA+PI4) RSS$( 1 )=ABS(TERM1+TERM2)-TAUCR KSS$(2 )=ABS(TERM1-TERM2)-TAUCR TERM3=-S0RT(2.)*RSB*SIN(THETA)*SIN(THETA+PI4)+ZSB o TERM4=RZS(I$1) "COS(THETA) RSS$(3)=ABS(TERM3+TERM4)-TAUCR RSS$(4)=ABS(TERM3-T ERM4)-TAUCR TERM5 = SQRT(2.)*RSB*C0S(THETA)*SIN(THETA+P14)-ZSB TERMG=RZS(I$1)*SIN(THETA) R5S$(5)=ABS(TERM5-TERM6)-TAUCR RS5$(6)=ABS(TERM5+TERM6)-TAUCR TERM7=S0RT(2.)*RSB*C0S(THETA)*C0S(THETA+PI4)-ZSB RSS$(7)=ABS(TERM7-TERM6)-TAUCR RSS$(8)=ABS(TERM7+TERM6)-TAUCR TERM8 = S0RT(2. ) *RSB*SIN(THETA ) *C0S(THETA + P14)+ ZSB RSS$(9)=ABS(TERM8+TERM4)-TAUCR RSS$(10)=ABS(TERM8-TERM4)-TAUCR CALCULATE MAXIMUN COMPONENT OF RSS RSMAX$=RSS$(1) IMAX$= 1 DO 30 I$=1 , 10 IF(RSMAX$.LT.RSS$(I$)) GO TO 20 GO TO 30 RSMAX$=RSS$(1$) IMAX$=I$ CONTINUE RSS(NV)=RSMAX$ IF(RSS(NV) LT.O.O) RSS(NV)=0.0 WRITE(6.41) NV.RSS(NV),IMAX$ NV=NV+1 CONTINUE 430 c 41 F0RMAT(5X,13,3X,F15.7,4X.13) C C CALCULATE RSSMAX AND RSSMIN C RSSMAX=RSS(1) C NV=NV-1 C C DO 50 1=1,NV 50 IF(RSSMAX.LT.RSS(I)) RSSMAX=RSS(I) C RSSMIN = RSS( 1 ) C DO 60 1=1,NV 60 IF(RSSMIN.GT.RSSO)) RSSMIN = RSS( I ) C C C CALCULATE NEW REGULAR GRID USING CGRID1 C NX = 50 NY = 50 IDIMX=NX XMIN=0.0 YMIN=0.0 XMAX=10. YMAX=10. DX=(XMAX-XMIN)/(NX-1) DY = (YMAX-YMIN)/(NY - 1) NRNG=2 C= 10. OPT=0.0 C CALL CGRID1(RSSG,IDIMX,NX,NY,XMIN.YMIN , 1 DX,DY,XP,YP,RSS,NV,C,NRNG,OPT) C C PLOT GRID C C XSIZE=10. YSIZE=10. NC = 10 IOP( 1 ) = 1 I0P(2)=1 IOP(3 )=0 I0P(4 ) = 2 I0P(5)=0 I0P(6)=1 I0P(7)=0 I0P(8)= 1 V0P(8)=.08 C DRSS=(RSSMAX-RSSMIN) K = 1 XNC=NC C C WRITE(6,70) 70 FORMAT(//,3X, 'I' ,1 1X, 'CVAL(I)' ,/) DO 80 1=1,NC XI = I XIN=(XI-1 . )/(XNC-1. ) CVAL(I)=RSSMIN+DRSS*XIN**K WRITE(6,81) I,CVAL(I) 80 CONTINUE 81 F0RMAT(2X,13,10X,F10.5) C CALL C0NTUR(XSIZE,YSIZE.RSSG,IDIMX,NX,NY, 1 CVAL.NC,IOP.VOP) C 431 CALL PLOTND RETURN END C C FUNCTION TO CALCULATE C C CRITICAL STRESSES AT T(I) C C FDR UNDOPED GAAS (M AND B) C REAL FUNCTION TAUCRM(DTA,TT) REAL DTA,TT C TAUCRM = G.8647E-2*EXP(5. 1989E3/(1511-DTA*(1.-TT))-3.4407) C RETURN END C C CRITICAL STRESSES FOR C C TE-DOPED GAAS (M AND B) C REAL FUNCTION TAUCTE(DTA,TT) REAL DTA,TT C TAUCTE=.39227*EXP(5.9205E3/(1511-DTA*(1.-TT))-3.9183) C RETURN END C C CRITICAL STRESSES C C FOR YIELD (S AND C) C C USED BY JORDAN C REAL FUNCTION TAUCRJ(DTA,TT) REAL DTA,TT C TAUCRJ=.5625*EXP(5.0596E3/(1511-DTA*( 1 .-TT))- 3.3485) C RETURN END 432 XI.7 Program for the Numerical Evaluation of Temperatures During Cooling c c C THIS PROGRAM CALCULATES THE C C TEMPERATURE FIELD DURING COOLING C C OF LEC-GROWN GAAS USING AN C C ANALYTICAL AXISYMMETRIC TIME C C DEPENDENT SOLUTION C C INTEGER NB,NG,NN,NR,NZ,IG REAL * 8 TIME,RO,ZT,DTA,H,HP RE AL *8 B,C,BETA,GAMMA REAL*8 FAC1 ,FAC2,FAC3,FAC4,FAC5 REAL*8 BF,CI,C2 REAL*8 TTIME,Z,RO,BARG,BFR REAL*8 TGAMMA,TBETA , TEM REAL*4 ARG.ARX REAL*4 TGR.CVAL.VOP.TMIN.TMAX REAL*4 RSIZE.ZSIZE REAL+8 DR.DZ.DTEM INTEGER IDIMR.NC.IOP C DIMENSION BETA(30),GAMMA(30),C1(30),C2(30) DIMENSION TEM(5000),TGR(11,41).IOP(8),V0P(8),CVAL(20) DIMENSION TBETA(50) C C READ CONSTANT AND PARAMETERS C C AND PRINT HEADLINES AND PARAMETERS C C READ(5,5) TIME 5 FORMAT( F 1 5 . 7 ) WRITE(6.6) 6 F0RMAT(//,5X,'ANALYTICAL SOLUTION FOR THE'./, 15X,'TRANSIENT TEMPERATURE FIELD DURING',/, 25X,'COOLING OF LEC-GROWN GAAS CRYSTALS'./) C READ(5,10) NB,NG READ(5,10) NR,NZ 10 F0RMAT(2I3) C READ(5,20) RO.ZT XL=ZT*RO 20 FORMAT(2F15.7) READ(5,20) DTA.H READ(5,20) B,C WRITE(6,30) TIME 30 FORMAT(5X,'TIME(NDIM)=',F15.7,/) C WRITE(6,40) RO.XL 40 FORMAT(5X,'RADIUS(CM)='.F10.5.5X, 1'LENGTH(CM)=',F10.5,/) WRITE(6,50) H,DTA 50 F0RMAT(5X,'HEAT TRANSFER COEFFICIENT(1/CM)=', 1F10.5./.5X,'DTA(DC)='.F10.5,/) WRITE(G,60)B.C 60 FORMAT(5X,'T(TIME=0)-TMP=',F9.3,'*Z+'.F7.3,'*Z**2',/) B=B/DTA C=C/DTA HP=H*R0 DO 80 1=1,NB READ(5,85) BETA(I) CONTINUE FORMAT(F15.7) DO 90 J=1,NG READ(5,85) GAMMA(J) GAMMA(U)=GAMMA( J ) / 4. CONTINUE CALCULATION OF COEFFICIENTS C 1 (BETA ) AND C2(GAMMA) WRITE(6,110) FORMAT(5X, ' BETA( I ) ',SX, 'CI(BETA)' ,//) DO 150 1=1.NB IF(BETA(I).LT.3.0) BF=B0FL3(BETA(I)) IF(BETA(I).GT.3.0.0R.BETA(I).E0.3.0) BF=B0FG3(BETA(I) ) C1(I)=(HP**2+BETA(I)**2)*BF C1(I)=1 ./C1(I) WRITE(G,20) BETA(I),C1(I) CONTINUE DO 200 d=1,NG ARG=GAMMA(J)*ZT FAC1 = 1 .+B*(ZT+HP/GAMMA(J)**2)+ 1 C*(ZT**2+2.*(ZT*HP-1.)/GAMMA(J)**2) FAC1=FAC1*SIN(ARG) FAC2=-HP+B*(1.-HP*ZT)+ 1 C*(2.*ZT-HP*ZT**2+2.*HP/GAMMA(J)**2) FAC2=FAC2*C0S(ARG)/GAMMA(J) FAC3=(HP-B-2.*C*HP/GAMMA(J)**2)/GAMMA(J) FAC4=ZT*(GAMMA(J)**2+HP**2)/2.+HP C2(J)=(FAC1+FAC2+FAC3)/FAC4 WRITE(6,20) GAMMA(J),C2(d) CONTINUE SUMMATION OF SERIES IN BESSEL FUNCTIONS WRITE(6,300) FORMAT( 10X, 'NN' ,6X, 'TEMP(ND)' ,6X, 'TEMP(* 10**3 DC)' ,//) DR=1 ./(NR- 1 . ) DO 400 1=1,NR R0 = DR*(I-1 ) TBETA(I)=0.0 DO 350 I$=1,NB ARX=-TIME*BETA(I$)**2 BARG=BETA(I$)*RO IF(BARG.LT.3.0) BFR=BOFL3(BARG) IF(BARG.GT.3.O.0R.BARG.E0.3.O) BFR=B0FG3(BARG) TBETA(I )=TBETA(I)+C1(I $)*EXP(ARX)*BFR CONTINUE CONTINUE SUMMATION OF SERIES IN FOURIER FUNCTIONS AND TEMPERATURE CALCULATION NN= 1 DZ=ZT/(NZ-1.) DO 500 d=1,NZ Z = DZ*(d- 1 ) TGAMMA = 0.0 DO 450 d$=1.NG ARG=GAMMA(d$)*Z ARX=-TIME*GAMMA(d$)**2 TGAMMA=TGAMMA+C2(J$)*EXP(ARX)* 1 (GAMMA(d$)*COS(ARG)+HP*SIN(ARG)) CONTINUE DO 480 I=1,NR TTIME=2.*HP*TBETA(I)*TGAMMA TGR( I , d ) = 1.238-(1.-TTIME)*DTA/1000.0 WRITE(6,510) NN,TTIME,TGR(I,d) NN=NN+1 CONTINUE CONTINUE FORMAT*10X,I5,6X,2F15.4) PLOT ISOTHERMS IDIMR=NR TMIN=TGR(1,1) DO 550 I=1,NR DO 550 d=1,NZ IF(TMIN.GT.TGR(I,d)) TMIN=TGR(I,d) CONTINUE 435 c WRITE(6,560) TMIN 5G0 FORMAT(//,'TMIN=',F10.6./) C TMAX=TGR(1.1) C DO 600 1=1,NR DO 600 J=1,NZ IF(TMAX.LT.TGR(I.J)) TMAX=TGR(I,J) 600 CONTINUE C WRITE(6.610) TMAX 610 FORMAT(/, 'TMAX =' ,F10.6,/) C RSIZE=3.0 ZSIZE = 12 .0 NC = 20 C IOP(1)=1 I0P(2)=1 I0P(3)=O I0P(4 ) = 1 I0P(5)=1 V0P(5)=3. I0P(6)=0 I0P(7)=O I0P(8)=1 V0P(8)=. 12 C DTEM=(TMAX-TMIN)/(NC-1.) C DO 700 I=1,NC CVAL(I)=TMAX-DTEM*(1-1.) WRITE(6,710) CVAL(I) 700 CONTINUE 710 F0RMAT(5X,F15.7) C C CALL CONTUR(RSIZE.ZSIZE.TGR,IDIMR,NR,NZ, 1CVAL,NC.IOP.VOP) C CALL PLOTND C STOP END C C C C POLYNOMIAL APPROXIMATION OF C C BESSEL FUNCTION FOR ARGUMENT C C LESS THAN 3.0 C C REAL FUNCTION BOFL3(DB) REAL*8 DB REAL BB BB=DB/3. C B0FL3=1,-2.2499997*BB**2+1.2656208*BB**4-10.3163866*BB**6+0.0444479*BB**8-20.0039444*BB**10+0.00021*BB**12 C RETURN END 436 c c C POLYNOMIAL APPROXIMATION OF C C BESSEL FUNCTION FOR ARGUMENT C C GREATER THAN 3.0 C C REAL FUNCTION B0FG3(DB) REAL*8 DB REAL CARG.BB C BB = 3 ./DB C B0FG3=.79788456-0.00000077*BB-0.0055274*BB**2-10.00009512*BB**3+0.00137237*BB**4-0.00072805*BB**5 2+0.00014476*BB**6 C CARG=3./BB-O.78539816-O.04166397*BB-10.00003954*BB**2+0.00262573*BB**3-0.00054125*BB**4 2-0.00029333*BB**5+0.00013558*BB**6 C B0FG3=B0FG3*C0S(CARG)*S0RT(BB/3.0) C RETURN END 437 APPENDIX XII XII.1 Program for the Numerical Evaluation of Analytical Axisymmetric Temperature Fields 1 0PEN4,4:CMD4 2 DIM AL(5) 3 FOR M=l TO 5 4 READ AL(M) 5 PRINT AL(M) 6 NEXT M 7 READ V,RO,K,ZT 8 PRINT "V=";V;"R0=";RO;"K=";K;"LENGTH=";ZT 9 FOR M=l TO 7 10 :LET Z=.25«M:PRINT CHRS(10);"HEIGHT=";Z;CHRS(10) 11 :FOR L=l TO 11 12 :LET RO=(L-l)«.l 13 LET TP=0 14 LET Kl=(V*R0)/(2*K) 15 REM LOOP TO SUM BESSEL SERIES 16 FOR N=l TO 5 17 :LET KA=SQR(K1~2+AL(N)-2) 18 :LET EZ=EXP(KA»Z) 19 :LET ET=EXP(-2*KA*ZT) 20 :LET C1=<EZ"(-1)-ET*EZ)/(1.-ET) 21 :LET X=ALCN)»RO 22 :IF X< = 3. THEN G0SUB 100 23 :IF X>3. THEN GOSUB 150 24 :LET JR=JX 25 :LET X=AL(N) 26 :IF X<=3. THEN GOSUB 100 27 :IF X>3. THEN GOSUB 150 28 :LET JA=JX 29 :LET TP=TP+<JR/JA)*<C1/<AL(N)"2+1.44)) 31 NEXT N 32 LET T=2.4*EXP(K1*Z)*TP 33 PRINT "RAD. POSITION3",RO;"TEMPERATURE=",T 36 :NEXT L 38 NEXT M 40 PRINT#4:CL0SE4 50 END 100 REM SUB BESSEL A 101 LET Jl=l.-2.245»(X/3)"2+1.26562*<X/3)"4-.31638*(X/3)"6+. 04444*(X/3)"8 102 LET J2=-.00394*<X/3)"10+.00024*(X/3)"12 103 LET JX=J1+J2 105 RETURN 438 150 REM SUB BESSEL B 151 LET Fl=.79788-.00552*<3/X>~2-.00009*(3/X>"3+.00137*(3/X) "4 152 LET F2=-.00072*<3/X)"5+.00014*(3/X)"6 153 LET FX=F1+F2 154 LET A1=X-.78539-.04166*(3/X)-.00004*(3/X>A2+.00262*(3/X) ~3 155 LET A2=-.00054*(3/X)^4-.00029*(3/X)"5+.00013*(3/X)"6 156 LET AX=A1+A2 157 LET JX=(FX*C0S(AX))/SQR(X) 159 RETURN 160 DATA 1.345,4.1225,7.1825,10.29,13.4125 161 DATA .001,2.,.04 162 DATA 2. 439 XII.2 Program for the Numerical Evaluation of Analytical Analytical Axisymmetric Stresses for Radial Temperature Fields 1 REM CALCULATION OF STRESSES FOR T=-R~2 2 DIM AN(50),BN(50) 4 READ NU,2T 6 0PEN4,4:CMD4:PRINT CHRS(27);CHRS(37);CHRS(67);"110" 7 PRINT CHRS(31);"CALCULATION OF STRESSES" 8 PRINT CHRS(30);CHRS(27);CHRS(48);"ANALYTICALLY" 13 PRINT "NU=";NU;"ZT=";ZT 14 PRINT "TEMPERATURE FIELD=-R~2" 20 REM CALCULATION OF AN AND BN 21 FOR N=l TO 10 22 :LET LN=(2*N-1)*3.1416/ZT 23 :LET T=LN/3.75 25 :IF T<=1. THEN GOSUB 180 26 :IF T>1. THEN GOSUB 190 27 :LET D=I0"2«LN-I1"2«(2»(1-NU)+LN"2)/LN 41 REM CAL ALN 42 : LET CA=D*(1-NU)*LN~5 43 : LET A1=I0*(4*NU+LN~2) 44 : LET A2=2*I1*(2*(1-NU)-NU*LN"2)/LN 46 : LET AN(N)=-2*(A1+A2)/CA 49 REM CALC OF BLN 51 : LET BN(N)=-2*(2*10-11*(4+LN~2)/LN)/CA 52 :PRINT "N="N;"AN(N)=";AN(N);"BN(N)=";BN(N) 53 NEXT N 55 REM CALCULATION OF - STRESSES 58 FOR L=3 TO 5 59 :LET Z=(L-2)*.5 60 :PRINT CHRS(IO) 62 IFOR 1=1 TO 11 63 PRINT CHRS(IO) "HEIGHT=";Z;CHRS(10) 64 : LET PR=0.0 65 : LET PT=0.0 66 : LET PZ=0.0 67 : LET PU=0.0 69 : LET RO=(I-l)*.l 72 : FOR N=l TO 10 73 : LET LN=(2*N-1)*3.1416/ZT 78 : IF R0=0.0 THEN GOTO 120 79 : LET T=LN*R0/3.75 80 : IF T<=1. THEN GOSUB 180 81 : IF T>1. THEN GOSUB 190 84 : LET Sl=(AN(N)*(10-11/(LN*R0))+BN(N)*(I1*LN*R0-(2*NU-1) *I0))*LN~3 85 : LET S3=(II*AN(N)/(LN*RO)-10*(2»NU-1)*BN(N))*LN"3 86 : LET S5=(-AN(N)*IO+BN(N)*((1-NU)*I1*LN*R0-2*(2-NU)*10)) *LN"3 440 87 ; 119 120 121 122 123 124 131 132 133 134 138 139 140 141 144 146 147 148 149 150 151 155 156 159 160 165 180 181 182 183 185 186 188 189 190 191 195 198 2«T" 199 200 201 202 205 220 LET S7= < AN < N)«I1+BN(N)*<I0«LN»R0+2»(1-NU)«I1>)»LN~3 : GOTO 131 REM STRESSES AT THE CENTER LET Sl=(AN(N)/2-BN(N>«(2«NU-l))»LN"3 LET S3=S1 LET S5=(-AN(N)-2»(2-NU)»BN(N))»LN"3 LET S7=0.0 LET S2=2»<2/LN-2+R0"2)/((1-NU)«LN> S4 = S2 S6=-8/(LN"3»(l-NU)) S8 = 4«R0/( (1-NU>»LN"2> PR=PR+(S1+S2)«SIN(LN»Z> PT=PT+(S3+S4)«SIN<LN»Z) PZ=PZ+(S5+S6)«SIN(LN«Z> PU=PU+(S7+S8)»COS<LN«Z> N=";N;"PR=";PR;"PT =";PT;"PZ=";PZ;"PU=";PU LET LET LET LET LET LET LET PRINT NEXT N LET SR=2»PR/ZT LET SH=2«PT/ZT LET SZ=2»PZ/ZT LET TR=2»PU/ZT PRINT "R0=";RO;"SR=";SR;"ST=";SH;"SZ=";SZ;"TR=";TR NEXT I NEXT L PRINT#4:CL0SE4 END RETURN REM CALCULATE 10 AND II FOR T<=1 LET P0=l+3.51562«T-2+3.08994«T~4+l.20674«T~6 LET Q0=.26597*T~8+.03607«T~10+.00458»T~12 LET I0=P0+Q0 LET Pl=.5+.8789»T-2+.51499»T~4+ . 15084*T~6 LET Ql= .02658«T~8+ . 00301 *T'"10+ .00032«T~12 LET 11=(P1+Q1)«T«3,75 RETURN REM CALCULATE 10 AND II FOR T>1 T=l/T P0=.39894+.01328*T+.00225»T~2-.00157»T"3 Q0=.00916»T~4-.02057*T~5+.02635«T~6-.01647«T~7+.0039 LET LET LET 8 LET LET LET LET I0=EXP(3.75/T)«(PO + CIO) »SQR(T/3.75> Pl= .39894- .03988*T- .00362»T~2+ .00163«T~3- . 01031 «T' Ql=.02282«T~5-.02895»T~6+.01787»T"7-.0042«T~8 I1=EXP(3.75/T>«(Pl+Ql)«SQR(T/3.75) RETURN DATA .29,2 44 1 XII.3 Program for the Numerical Evaluation of Analytical Axisymmetric Stresses for Axisymmetric Temperature Fields 1 REM CALCULATION OF SIGR 2 DIM AL(5) 4 READ H,RO,K,V,NU,ZT 5 LET CT=(V*RO)/(2*K) 6 0PEN4,4:CMD4:PRINT CHRSC27);CHRS<37);CHRS(67);"120" 7 PRINT CHRSOl);"CALCULATION OF RADIAL STRESSES" 8 PRINT CHRSC30);CHRS(27);CHRS(48);"ANALYTICALLY" 9 FOR M=l TO 5 10 :READ AL(M) 11 :PRINT AL(M) 12 NEXT M 13 PRINT "H=";H;"RO=";RO;"V=";V;"K=";K;"NU="; NU;"ZT=";ZT 15 FOR L=3 TO 5 16 LET Z=(ZT*.1)"2*2~(L-1) 17 PRINT CHRS(10);"HEIGHT=";Z;CHRS(10) 18 FOR 1=1 TO 11 19 LET SP=0.0 20 LET RO=(I-l)*.l 21 FOR N=l TO 5 22 :LET LN=N*3.1416/ZT 23 :LET T=LN/3.75 25 :IF T<=1. THEN GOSUB 100 26 :IF T>1. THEN GOSUB 150 27 :LET D=I0~2«LN-I1~2*(2*(1-NU)+LN~2)/LN 31 REM CALC OF KT 32 :LET KT=0 33 :FOR M=l TO 5 35 :GOSUB 180 37 .'LET KT=KT+ZA/(CZ*AL(M)~2) 39 : NEXT M 41 REM CAL ALN 43 :LET A1=I0»(LN~2+2*H*R0*(1-NU)) 44 :LET A2=I1*(LN"2*(H*R0+2*(1-NU))+2*H*R0*(l -NU))/LN 46 :LET AN=2*H*R0*(Al+A2)*KT/(D*(1-NU)*LN~2) 49 REM CALC OF BLN 50 :LET CB=-2*(H*R0)"2/(D»(1-NU)«LN"2) 51 :LET B=I0+I1«LN/(H«RO) 52 :LET BN=CB*B«KT 55 REM CALC SRR 58 :IF R0=0 THEN GO TO 83 59 :LET T=LN»R0/3.75 60 :IF T<=1. THEN GOSUB 100 61 :IF T>1. THEN GOSUB 150 63 :LET Sl=(10*(AN-BN*(2«NU-1))+11*(BN*LN*RO-AN/(LN«RO)))*LN o 65 REM CALC OF KTJ 66 :LET S2=0 67 :FOR M=l TO 5 68 : GOSUB 180 442 69 : LET X=AL(M)»RO 70 : IF X<=3. THEN GOSUB 200 71 : IF X>3. THEN GOSUB 250 72 : LET J2=J0«(LN/AL(M)>~2 73 : LET J3=J1/X 74 : LET X=AL(M) 75 : IF X< = 3. THEN GOSUB 200 76 : IF X>3. THEN GOSUB 250 77 : LET J4=ZA/(JO«CZ> 78 : LET S2=S2+J4«<J3+J2) 79 !NEXT M 81 :LET SP=SP+(S1-2«H»R0»S2/(l-NU)>«SIN(LN»Z) 82 IGOTO 93 83 :LET Sl=(AN/2-BN*<2»NU-1))»LN~3 84 :LET KJ=0 85 :F0R M = l TO 5 86 : GOSUB 180 87 : LET X=AL(M) 88 : IF X<=3. THEN GOSUB 200 89 IF X>3. THEN GOSUB 250 90 : LET KJ=KJ+ZA«((LN/KA)~2+.5)/<JO«CZ) 91 '.NEXT M 92 :LET SP=SP+<S1-2»H«R0»KJ/(l-NU)>«SIN(LN«Z> 93 : PRINT ,,N = ";N; "SP=M;SP 94 NEXT N 95 LET SR=2»SP/ZT 96 PRINT"RO=";RO; "SR=,,;SR 97 NEXT I 98 NEXT L 99 PRINT#4:CL0SE4:END 100 REM CALCULATE 10 AND II FOR T<=1 101 LET P0=l+3.51562»T-2+3.08994«T~4+l.20674«T~6 102 LET 00=.26597»T~8+.03607«T~10+.00458»T~12 103 LET IO=PO+QO 105 LET Pl=.5+.8789«T"2+.51499«T~4+.15084*T~6 106 LET 01=.02658*T~8+.00301»T~10+.00032»T~12 108 LET I1=(P1+Q1)»T*3.75 110 RETURN 150 REM CALCULATE 10 AND II FOR T>1 151 LET T=l/T 155 LET P0=.39894+.01328»T+. 00225«T~2-.00157»T~3 158 LET Q0=.00916«T~4-.02057«T~5+.02635«T~6-.01647«T~7+.0039 2«T~8 159 LET I0=EXP(3.75/T>«(PO+QO)*SQR(T/3.75) 160 LET Pl=.39894-.03988«T-.00362*T~2+.00163«T"3-.01031»T~4 161 LET 01=.02282«T"5-.02895«T~6+.01787«T~7-.0042»TA8 162 LET I1=EXP(3.75/T)*(Pl+Ql)«SQR(T/3.75) 165 RETURN 180 REM CALCULATION OF ZA AND CZ 181 LET KA=SQR(AL(M)"2+CT-2) 182 LET ET=EXP(-KA«ZT) 183 LET EV=EXP(CT»ZT) 184 LET CZ=(AL(M > ~2+(H»RO)~2)«<l+<LN/AL(M))~2)»(1-ET~2) 187 LET Zl= <1/ET-(-1)~N*EV)/((CT-KA)~2+LN"2) 188 LET Z2=(ET-(-l)~N«EV)/( (CT+KA)~2+LN~2> 443 189 LET ZA=LN*ET*(Z1-Z2) 193 RETURN 200 REM CALCULATION OF JO AND Jl FOR X<=3 201 LET Y=X/3 202 LET G1=1-2.245*Y~2+1.26562*Y~4-.31638*Y~6 203 LET G2=.04445*Y~8-.00394*Y~10+.00021»Y~12 204 LET J0=G1+G2 206 LET G3=.5-.56249*Y~2+.21093«Y~4 207 LET G4=.03954*Y~6+.00443*Y~8-.00031*Y~10 208 LET J1=(G3+G4)»X 210 RETURN 250 REM CALCULATION OF JO AND Jl FOR X>3 251 LET Y=3/X 252 LET Fl= .79788-.00552*Y-"2-.00009*Y~3+.00137*Y~4 253 LET F2=-.00072*Y~5+.00014»Y"6 254 LET T1=X-.78539-.04166*Y-.00004*Y~2+.00262*Y~3 255 LET T2=-.00054*Y~4-.00029*Y~5+.00013*Y"6 256 LET J0=(F1+F2)*C0S(T1+T2)/SQR<X) 258 LET F3=.79788+.01659*Y~2+.00017*Y~3-.00249*Y~4 259 LET F4=.00113*Y"5-.0002*Y~6 260 LET T3=X-2.35619+.12499*Y+.00005*Y~2-.00638*Y~3 261 LET T4=.00074»Y~4+.00079*Y~5-.00029*Y~6 264 LET Jl=(F3+F4)*C0S(T3+T4)/SOR(X) 266 RETURN 270 DATA .6,2.,.04,.001,.29 271 DATA 5 272 DATA 1.345,4.1225,7.1825,10.29,13.4125 444 APPENDIX XIII TEMPERATURE AND STRESS PLOTS FROM THE COMPUTER PROGRAMS OUTPUT (GROWTH) Units are : 103°C for the temperatures and MPa for the stresses. XI11 . 1 Cone Angle CL, 10 mm ; R, 20 mm : B, 10 mm ; AP, 30 atm. BG, 100 C/cm ; AG, 50 C/cm. XI11.1.1 Cone Angle 7.1° XIII.1.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.1.1.2 Cross-sections at 2.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) XIII.1.1.3 Cross-sections at 7.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) XI11.1.2 Cone Angle 30° XIII.1.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.1.2.2 Cross-sections at 7.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; XIII.1.3 Cone Angle 45° 445 XIII.1.3.1 (a) Temperature and Von Nlses Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) ; (f) Slip Mode. XIII.1.3.2 Cross-sections at 2.5 mm from the interface (a) MRSS-CRSS (Yield) ; (b) MRSS-CRSS (MBTe) XIII.1.3.3 Cross-sections at 7.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MBTe) XIII.1.4 Cone Angle 54.7° XIII.1.4.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) XIII.1.4.2 Cross-sections at 2.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; XIII.1.4.3 Cross-sections at 7.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; XI 11.1.5 Cone Angle 65° XIII.1.5.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.1.5.2 Cross-sections at 8.0 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; XIII.1.6 Effect of Thermal Conditions XIII.1.6.1 BG, 50°C/cm ; AG, 25°C/cm ; CA, 45° . XIII.1.6.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.1.6.1.2 Cross-sections at 5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) XIII.1.6.1.3 Cross-sections at 7.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) XIII.1.6.2 BG, 200°C/cm ; AG, 100°C/cm 446 XIII.1.6.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MBTe) . XIII.1.6.2.2 Cross-section at 2.5 mm from the interface, MRSS XIII.1.6.2.3 Cross-sections at 7.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; XIII.1.6.3 BG, 400°C/cm ; AG, 100°C/cm XIII.1.6.3.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) Slip Mode. XIII.1.6.3.2 Cross-sections at 7.5 mm from the interface (a) MRSS-CRSS (Yield) ; (b) Slip Mode. XIII.1.7 Heat Transfer Coefficient BG, 100°C/cm ; AG, 50°C/cm XI 11.1.7.1 H/l . 3 XIII.1.7.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.1.7.1.2 MRSS-CRSS (Yield) in cross-sections at distance from the interface (a) 2.5 mm ; (b) 7.5 mm. XIII.1.7.2 Hxl.3 XIII.1.7.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.1.7.2.2 MRSS-CRSS (Yield) in cross-sections at 2.5 mm from the interface XIII.1.7.2.3 Cross-sections at 7.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (a) Teaperature and Von Mises Stress <b> MRSS : (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) co o o o o 453 XIII.1.1.2 Cross-sections at 2.5 m from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) 454 455 456 457 XIII.1.1.3 Cross-sections at 7.5 ma froi the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) 459 XIII.1.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 461 a 462 463 c XIII.1.2.2 Cross-sections at 7.5 mm fron the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; 465 466 XIII.1.3.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 468 a 469 XIII.1.3.2 Cross-sections at 2.5 urn from the interface (a) MRSS-CRSS (Yield) ; (b) MRSS-CRSS (MBTe) 470 471 b 472 XIII.1.3.3 Cross-sections at 7.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MBTe) 473 474 477 478 479 480 XIII.1.4.2 Cross-sections at 2.5 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; 481 a 482 XIII.1.4.3 Cross-sections at 7.5 urn from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; 484 485 XIII.1.5.1 (a) Teaperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 486 487 XIII.1.5.2 Cross-sections at 8.0 mn fro» the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; MRSS - CRSS (Y) , M Pa 490 MRSS-CRSS(MB).MPa MRSS-CRSS(Te), MPa XIII.1.6.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 491 MRSS.MPo HEIGHT=0.5cm a [0103 492 t 493 MRSS-CRSS(MB)-MPo H E I G H T = 0. 50 c m C CO I 0] 494 XIII.1.6.1.2 Cross-sections at 5 mm fro» the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) 495 MRSS, M Po HEIGHT = 0.75cm [OIO] 496 MRSS - CRSS (Y ), MPa HE I GHT = 0.7 5 cm b CO 10] 497 MRSS-CRSS(MB),MPo HEIGHT = 0.75cm C CO I 0] 498 MRSS-CR SS(Te), MPo HE I G H T = 0.75cm d CO I 0] XIII.1.6.1.3 Cross-sections at 7.5 froa the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) 500 MRSS,MPa MRSS - CRSS (Y) , M Pa 501 MRSS - CRSS (Te) , MPa d XIII.1.6.2.1 (a) Teaperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MBTe) . 502 XIII.1.6.2.2 Cross-section at 2.5 mm fron the interface, MRSS 503 504 MRSS-CRSS( Y) , MPa b H E I G H T = 0.7 5 c m CO 10] XIII.1.6.2.3 Cross-sections at 7.5 an from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; 506 b c 507 o-o-o-o-o-o 6 o A A & • p044ill I I I O 60 A A • 6 O O O D o • I V o 000 000^ 6000000 VA 600000 oso 6000000 o\> OOOOOOOOOO^ OOOOOOOOOOO 0000000000 o\> 1 \ 000 OOOOOOOOO ON 600 OOOOOOOOOOO OOOOOOAAAAAAAAO. 1 \ OOOOAAAAAA A A A & A • OOAAAAAAAOOOOOD o^d O O O O O A A AAAAAAAAAO O 6 O D O • • A AOOOOOOOOO OOOOOOQAAOOOOOOO I I OODODOOAAOOOOOOOO O D • • A A OOOOODo6 6 6 ? ODOOODOAAOOOOOO OODOODODDOAAOOOOA OO-ODODOODOOOAAAOO -A-A-A-A- A- A-A-O-O-O- O-O-O 6-A-A-^ d XIII.1-6.3.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) Slip Mode. 508 MRSS - CRSS ( Y ), MPa HE I G H T = 0.75 cm a CO I 0] 509 O ( o ( D ( A ( • ( [1101 LOT I ] 10 I U L I OT] [T0T3 b tO 10] A a e MOT] I I OT] tOII] [Oil] [7 I 0] [110] t I T 0] A a a / S » * A a / - - a a a a • a o • • • A A / °*» a o • • o A A ° o a .A* A o a a a a a o o o o a a a A A a o o o o o o • • e o o o • • • N a a a. • • » _ \ • • » 8 a 8 \ a a o a a a • a - - ^ « a a a a a a • • • % D 9*«aaaaaaa • noDDOAAAAaao o • o a a • o I a I 4 GaaceoaAAAAooo 999909OAAA A A • \ 9 9 9 9 9 & * A 9 V • • ^ ^ A A 9 9 A 9 • 9 A 9 9 9 O • • • • a lj w ~ y 9 ••••••oaaSBBB * ••••^•••oooaaaaa * I 90D0DAa000000999 [1001 * _ _4 •®9©0«B00009 9 9« A "••oaaao999999 A A A A 'A. _ °B999999 ® 0 •••999« « 0 ® ® 9 * 9 • ^ 9 9 * 9 • "99 • a / XIII.1.6.3.2 Cross-sections at 7.5 mm from the interface (a) MRSS-CRSS (Yield) ; (b) Slip Mode. 511 MRSS - CRSS (MB), MPa MRSS - CRSS (Te) , MPa d e XIII.1.7.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 513 MRSS-C RSS ( Y) , M Po HEIGHT =0. 25cm a CO 10] 514 MRSS - CRSS ( Y ), MPo HE I GHT = 0.75cm b CO I 0] C I 003 XIII.1.7.1.2 MRSS-CRSS (Yield) in cross-sections at distance fron the interface (a) 2.5 nm ; (b) 7.5 mm. MRSS.MPa MRSS-CRSS (Y) , MPa MRSS-CRSS(MB),MPa MRSS-CRSS(Te), MPa 518 MRSS - CRSS ( Y ), MPa HEIGHT =0. 25cm CO I 0] II.1.7.2.2 MRSS-CRSS (Yield) in cross-sections at 2.5 an from the interface 519 MRSS, M Po HElGHT = 0.7 5cm a [0-103 520 II.1.7.2.3 Cross-sections at 7.5 mm fro« the interface (a) MRSS ; (b) MRSS-CRSS (Yield) ; 521 XIII.2 Crystal Length R, 27.5 mm ; B.21 mm ; AP, 30 atm. ; CA, 30° . XI I I .2. 1 CL, 13.75 mm XIII.2.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.2.1.2 Cross-sections (a) MRSS-CRSS (Yield) at 5.5 from the interface ; (b) MRSS at 11.0 mm from the interface ; (c) MRSS-CRSS (Yield) at 11.0 mm from the interface XI I I.2.2 CL, 27.5 mm XIII.2.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.2.2.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of : (a) 5.5 mm ; (b) 11.0 mm ; (c) 19.25 mm ; (d) 24.75 mm ; (e) 27.5 mm . XI I I.2.3 CL, 55.0 mm XIII.2.3.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.2.3.2 MRSS-CRSS (Yield) at 5.5 mm from the interface. XIII.2.3.3 Cross-section at 11.0 mm from the interface (a) MRSS ; (B) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe). XIII.2.3.4 MRSS-CRSS (Yield) at a distance from the interface of (a) 19.25 mm ; (b) 24.75 mm XIII.2.3.5 Cross-section at 33 mm from the interface (a) MRSS ; (B) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe). XIII.2.3.6 MRSS-CRSS (Yield) at a distance from the interface of (a) 38.5 mm ; (b) 46.75 mm ; (c) 52.25 mm. XI11.2.4 CL, 82.5 mm 522 XIII.2.4.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.2.4.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 5.5 mm ; (b) 11.0 mm ; (c) 19.25 mm ; (d) 24.75 mm ; (e) 33.0 mm ; (f) 38 .5 mm ; (g) 46.75 mm ; (h) 52.25 mm ; (i) 60.5 mm ; (j) 66 mm ; (k) 74.25 mm ; (1) 79.75 mm . XII I . 2.5 CL, 110 mm XIII.2.5.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.2.5.2 MRSS-CRSS (Yield) in cro from the interface of (a) 5.5 mm; (b)11.0 (d) 33.0 mm ; (e) 96. ss-sections at a distance mm ; (c) 24.75mm ; 2 5 mm 526 a 527 528 c XIII.2.1.2 Cross-sections (a) MRSS-CRSS (Yield) at 5.5 from the interface ; (b) MRSS at 11.0 mm from the interface ; (c) MRSS-CRSS (Yield) at 11.0 mm from the interface. 530 XIII.2.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 531 i a 532 533 c 534 535 e XIII.2.2.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 5.5 am ; (b) 11.0 mm ; (c) 19.25 am ; (d) 24.75 am ; (e) 27.5 mm . 536 XI11.2.3.1 (a) (b) (d) Tenperature and Von Mises Stress MRSS ; (c) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) w ~3 538 XIII.2.3.2 MRSS-CRSS (Yield) at 5.5 mm from the interface 539 540 541 542 XI I I .2.3 3 Cross-section at 11.0 BID fron the interface (a) MRSS ; (B) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) 543 544 XIII.2.3.4 MRSS-CRSS (Yield) at a distance from the interface of (a) 19.25 mm ; (b) 24.75 mm 545 546 547 o c 548 XIII.2.3.5 Cross-section at 33 mm from the interface (a) MRSS ; (B) MRSS-CRSS (Yield) ; (c) MRSS-CRSS (MB) ; (d) MRSS-CRSS (MBTe) 549 550 551 XIII.2.3.6 MRSS-CRSS (Yield) at a distance from the interface of (a) 38.5 urn ; (b) 46.75 mm ; (c) 52.25 BB. 553 554 XIII.2.4.1 (a) Te»perature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 555 a 556 557 c 558 d 559 560 561 562 i 564 565 566 XIII.2.4.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 5.5 Rn ; (b) 11.0 mm ; (c) 19.25 mm ; (d) 24.75 mm ; (e) 33.0 mm ; (f) 38 .5 mm ; (g) 46.75 mm ; (h) 52.25 mm ; (i) 60.5 mm ; (j) 66 mm ; (k) 74.25 mm ; (1) 79.75 mm . a 567 569 570 57 1 572 573 574 e XIII.2.5.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of (a) 5.5 mm ; (b) 11.0 mm ; (c) 24.75 mm ; ( d) 33 . 0 mm ; (e) 96.25 ma . 575 XI11.3 Crystal Radius R, 40 mm ; B, 21 mm ; CA, 30° ; AP, 30 atm. XI11.3.1 CL, 20 mm XIII.3.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.3.1.2 Cross-sections at 8 mm from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) XIII.3.1.3 Cross-sections at 16 mm from the interface (a) MRSS ; (B) MRSS-CRSS (Yield) XI I I.3.2 CL, 40 mm XIII.3.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) . XIII.3.2.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 8 mm ; (b) 16 mm ; (c) 28 mm ; (d) 36 mm . XI I I .3.3 CL, 80 mm XIII.3.3.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.3.3.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 8 mm ; (b) 16 mm ; (c) 28 mm ; (d) 36 mm ; (e) 48 mm ; (f) 56 mm ; (g) 68 mm ; (h) 76 mm . XI11 . 3 . 4 CL, 100 mm (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XI I I . 3.5 CL, 120 mm XIII.3.5.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.3.5.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 8 mm ; (b) 16 mm ; (c) 28 mm ; (d) 36 mm ; (e) 48 mm ; (f) 56 mm ; (g) 68 mm ; (h) 76 mm ; (i) 88 mm ; (j) 96 mm ; (k) 108 mm ; (1) 116 mm . 577 XI 11 .3 . 1 . 1 (a) (b) (d) Temperature and Von Mises Stress MRSS ; (c) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 578 579 XIII.3.1.2 Cross-sections at 8 am from the interface (a) MRSS ; (b) MRSS-CRSS (Yield) 580 581 b XI I I .3. 1.3 Cross-sections at (a) MRSS ; (B) 16 nm from the interface MRSS-CRSS (Yield) 583 XIII.3.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) 584 a 585 b 586 e 587 XIII.3.2.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 8 in ; (b) 16 mm ; (c) 28 am ; (d) 36 am . o 01 oo CO 590 XI I I .3.3.1 (a) (b) (d) Temperature and Von Mises Stress MRSS ; (c) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 591 592 593 c 594 595 596 597 598 XIII.3.3.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 8 mm ; (b) 16 mm ; (c) 28 mm ; (d) 36 mm ; (e) 48 ma ; (f) 56 mm ; (g) 68 mm ; (h) 76 mm . 599 CL, 100 BB (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 602 £09 604 XIII.3.5.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 605 606 607 608 609 610 611 612 614 615 616 XIII.3.5.2 MRSS-CRSS (Yield) in cross-sections at a distance of the interface of (a) 8 Bm ; (b) 16 BB ; (c) 28 mm ; (d) 36 ii ; (e) 48 BB ; (f) 56 am ; (g) 68 BB ; (h) 76 BB ; (1) 88 Bm ; (j) 96 BB ; (k) 108 BB ; (1) 116 BB . 617 XI 11.4 Growth Velocity R 27.5 mm ; CL, 55 mm ; CA, 30° ; B, 21 mm XIII.4.1 V, 0.0001 cm/s (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.4.2 V, 0.01 cm/s (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 618 620 d e XIII.4.1 V, 0.0001 cu/a (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 621 a 623 o XIII.4.2 V, 0.01 cm/s (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 624 XIII.5 Thermal Condition R, 27.5 mm CL , 55 mm ; CA, 30 ; AP, 30 atm XIII.5.1 B , 21 mm XIII.5.1.1 G/2 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I .5. 1 .2 G/4 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I .5. 1 .3 G/6 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.5. 1.4 G/8 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I . 5 . 2 B, 4 0 mm XI11 .5.2 . 1 G (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.5.2.2 G/2 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XII I . 5.2 . 3 G/3 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I . 5.3 B, 50 mm XIII .5.3.1 G (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 625 XI I I.5.3.2 G/2 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I.5.3.3 G/3 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 626 a 628 XI 11 .5.1 .1 G/2 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 629 630 631 XI I I .5.1 .2 G/4 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) a 633 634 d e XI I I .5. 1 . 3 G/6 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 635 a 637 XI I I .5.1 . 4 G/8 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 638 639 Temperature and Von Mises Stress MRSS ; (c) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 64 1 642 643 d e XI I I.5.2.2 G/2 (a) Teuperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 645 646 d e XI I I .5.2.3 G/3 (a) Teuperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 647 649 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 650 a XI I I .5.3.2 G/2 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; a 653 654 XI I I.5.3.3 G/3 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 655 XIII.6 Thermal Conditions R, 40 mm CL, 80 mm ; CA, 30° ; AP, 30 atm. XIII.6.1 B,21 mm XI 11.6.1.1 G/4 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I .6.1 .2 G/8 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I.6.2 B, 40 mm XI11.6.2.1 G (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.6.2.2 G/2 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XIII.6.2.3 G/3 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I . 6.2.4 G/5 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I . 6 . 3 B, 50 mm XI I I .6 . 3 . 1 G XI 11 . 6 . 3 . 1 . 1 CL, 60 ram (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) XI I I.6.3.1.2 CL, 80 mm (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 656 XIII.6.3.2 G/3 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I . 6 . 3.3 G/5 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 658 659 e G/4 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 660 662 XI I I .6.1.2 G/8 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 663 a 665 d e XIII.6.2.1 G (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 666 667 668 669 670 671 3 If > XI I I .6.2.3 G/3 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 672 674 1.042 1.066 1.091 1.115 1.140 1.164 1.189 1.213 676 677 679 680 XI 11.6.3.1. 1 CL, 60 mm (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 681 a 682 683 XI I I .6.3.1.2 CL, 80 mm (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 684 a 685 686 XI I I .6.3.2 G/3 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 687 a 690 XIII.7 Argon Pressure, 2 atmospheres. R, 27.5 mm ; CL, 55 mm ; CA, 30° ; B, 21 mm XI I I . 7 . 1 G (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I . 7 . 2 G/2 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I .7.3 G/4 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) XI I I .7 . 4 G/6 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) a 693 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 696 XI I I .7.2 G/2 (a) Tenperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) G/4 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS 700 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBT 703 XIII.8 Curvature of the Solid/Liquid Interface CA, 30 ; AP, 30 atm. XIII.8.1 Convex Interface XIII.8.1.1 B, 21 mm ; G. XIII.8.1.1.1 R, 27.5 mm XI 11.8.1.1 . 1 . 1 CL, 27.5 mm (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XI I I . 8 . 1 . 1. 1 .2 CL, 55 mm XIII.8.1.1.1.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XI I I.8.1.1.1.2.2 MRSS-CRSS (Yield) in cross-sections at distances from the interface of (a) 11 mm ; (b) 24.75 mm ; (c) 38.5 mm ; (d) 52.25 mm ; XI I I .8.1 . 1 . 1 .3 CL, 82.5 mm XI11.8.1.1 . 1 . 3 . 1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . X I I I.8.1.1.1.3.2 MRSS-CRSS (Yield) in cross-sections at (along the axis) distances from the interface of (a) 11 mm ; (b) 24.75 mm ; (c) 38.5 mm ; ( d ) 52.5 mm ; (e) 66 mm . XIII.8.1.1.2 R, 40.0 mm ; B, 21 mm ; G ; CL, 80 mm . XI11.8.1 . 1 . 2 . 1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XI I I.8.1.1.2.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) from the interface of (a) 16 mm ; (b) 36 mm ; (c) 56 mm . XIII.8.1.2 B, 50 mm ; R, 2.75 mm XIII.8.1.2.1 G X111.8.1.2.1 . 1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 704 XI I I . 8 . 1 . 2. 1 .2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) from the interface of (a) 11 mm ; (b) 36 mm ; (c) 56 mm ; (d) 76 mm XIII.8.1.2.2. G/3 XIII.8.1.2.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . X I I I . 8 . 1 . 2 . 2.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) from the interface of (a) 11 mm ; (b) 36 mm ; (c) 56 mm ; (d) 76 mm XIII.8.2 Concave Interface B,21 mm ; G . XIII.8.2.1 R, 27.5 mm ; CL, 55 mm . XIII.8.2.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.8.2.1.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) from the interface of (a) 2.75 mm ; (b) 16.5 mm ; (c) 30.25 mm ; (d ) 4 4 mm XIII.8.2.2 R, 40 mm ; CL.80 mm . XIII.8.2.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) . XIII.8.2.2.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) from the interface of (a) 2.75 mm ; (b) 16.5 mm ; (c) 30.25 mm ; (d ) 44 mm 705 706 707 708 709 710 XIII.8.1.1.1.1 CL, 27.5 mm (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 711 a 712 713 XIII.8.1.1.12.1 (a) (b) (d) Temperature and Von Mises Stress MRSS ; (c) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 714 a 715 716 717 d X I I I . 8 . 1.1.1.2.2 MRSS-CRSS (Yield) in cross-sections at distances from the interface of (a) 11 am ; (b) 24.75 am ; (c) 38.5 am (d) 52.25 am ; 718 a 720 XIII.8.1.1.1.3.1 (a) Tenperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 721 722 723 724 725 X111.8.1.1.1.3.2 MRSS-CRSS (Yield) in cross-sections at (along the axis) distances from the interface of (a) 11 aa ; (b) 24.75 am ; (c) 38.5 an ; (d) 52.5 aa ; (e) 66 am . 726 727 728 XIII.8.1.1.2.1 (a) (b) (d) Teuperature and Von Mises Stress MRSS ; (c) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 729 730 731 c 732 XII I.8.1.1.2.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) fro» the interface of (a) 16 ma ; (b) 36 mm ; (c) 56 mm . 733 a 735 XIII.8.1.2.1.1 (a) Temperature and Von Mises Stress (b) MRS S ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 736 a 737 738 739 XI I I.8.1.2.1.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) from the interface of (a) 11 mm ; (b) 36 mm ; (c) 56 mm ; (d) 76 mm 742 XIII.8.1.2.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 743 744 745 746 XI 11.8.1.2.2.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) from the interface of (a) 11 mm ; (b) 36 mm ; (c) 56 mm ; (d) 76 mm. a 748 749 XIII.8.2.1.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 750 751 752 753 d 8.2.1.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) from the interface of (a) 2.75 BB) ; (b) 16.5 BB ; (c) 30.25 BB ; (d) 44 BB . 754 755 756 XIII.8.2.2.1 (a) Temperature and Von Mises Stress (b) MRSS ; (c) MRSS-CRSS (Yield) ; (d) MRSS-CRSS (MB) ; (e) MRSS-CRSS (MBTe) 757 758 759 760 d XIII.8.2.2.2 MRSS-CRSS (Yield) in cross-sections at distances (along the axis) fron the interface of (a) 2.75 an ; (b) 16.5 BB ; (c) 30.25 an ; (d) 44 BB . 761 APPENDIX XIV TEMPERATURE AND STRESS PLOTS FROM THE COMPUTER PROGRAMS OUTPUT (COOLING) Units are, 103oC for the temperatures and MPa for the stresses. R, 27.5 mm ; CL, 110 mm . XIV.1 Ambient Temperature 800°C, Argon XIV.1.1 Initial Gradient GC XIV.1.1.1 Time 10 seconds XIV.1.1.1.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.1.1.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 25.5 mm . XIV.1.1.1 Time 10 seconds XIV.1.1.1.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.1.1.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.1.1.2 Time 20 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.1.1.3 Time 60 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . 762 XIV.1.1.4 Time 300 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) XIV.1.2 Initial Gradient GC/2 ; Time 10 seconds . XIV.1.2.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.1.2.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.1.3 Initial Gradient GC/4 XIV.1.3.1 Time 10 seconds XIV.1.3.1.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) . XIV.1.3.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.1.3.2 Time 20 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MBTe) . XIV.1.3.3 Time 60 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . 764 XIV. 1 . 1 . 1 1 (a) (c) (e) Temperature ; (b) Von Mises Stress MRSS ; (d) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 766 767 768 769 d XIV.1.1.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 25.5 mm . 772 e f XIV.1.1.2 Tiae 20 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 775 XIV.1.1.3 Time 60 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 778 e f XIV.1.1.4 Time 300 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) c 780 d XIV.1.2.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 782 783 784 785 XIV.1.2.2 MRSS-CRSS (Yield) in cross-sections at a distance froa the interface of : (a) 2.75 an ; (b) 8.25 BID ; (c) 16.5 mm ; (d) 2 7.5 mm . 788 XIV.1.3.1.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) . 789 790 b 791 c 792 d XIV.1.3.1.2 MRSS-CRSS (Yield) in cross-sections at a distance frou the interface of (a) 2.75 BB ; (b) 8.25 BB ; (c) 16.5 BB ; (d) 27.5 BB . e 795 XIV.1.3.2 Time 20 seconds (a) Temperature ; (b) Von Nises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MBTe) . 798 e f XIV.1.3.3 Tiae 60 seconds (a) Teuperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 799 XIV.2 Ambient Temperature 1000°C XIV.2.1 Boron Oxide XIV.2.1.1 Initial Gradient GC XIV.2.1.1.1 Time 5 seconds XIV.2.1.1 . 1 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.1.1.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.1.1.2 Time 10 seconds XIV.2.1 . 1 . 2 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.1.1.2.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.1.1.3 Time 20 seconds XIV.2.1 . 1 . 3 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.1.1.3.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27 . 5 mm . XIV.2.1.1.4 Time 60 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.1.2 Initial Gradient GC/2 ; Time 10 seconds . (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.1.3 Initial Gradient GC/4 XIV.2.1.3.1 Time 10 seconds 800 XIV.2.1.3.1.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.1.3.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.1.3.2 Time 20 seconds XIV.2.1.3.2 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.1.3.2.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.2 Argon XIV.2.2.1 Initial Gradient GC XIV.2.2.1.1 Time 5 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.2.1.2 Time 10 seconds XIV.2.2.1.2.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.2.1.2 . 2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.2.1.3 Time 20 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.2.1.4 Time 60 seconds XIV.2.2.1.4.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . 801 XIV.2.2.1 . 4 . 2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.2.1.5 Time 300 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (MB) . XIV.2.2.2 Initial Gradient GC/2 XIV.2.2.2.1 Time 10 seconds XIV.2.2.2 . 1 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.2.2.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.2.2.2 Time 20 seconds XIV.2.2.2.2.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.2.2.2.2 MRSS-CRSS (Yield) in cross-sect1ons at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d ) 27.5 mm . XIV.2.2.2.3 Time 60 seconds XIV.2.2.2 . 3.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.2.2.3.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.2.3 Initial Gradient GC/4 XIV.2.2.3.1 Time 10 seconds XIV.2.2.3. 1 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 802 XIV.2.2.3.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . XIV.2.2.3.2 Time 60 seconds XIV.2.2.3.2.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) . XIV.2.2.3.2.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . 805 XIV.2.1.1.1 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 806 807 808 809 d XIV.2.1.1 . 1 . 2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of (a) 2.75 am ; (b) 8.25 mm ; (c) 16.5 in ; (d) 27.5 am . 812 XIV.2.1 . 1 . 2 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 813 814 815 816 XIV.2.1.1.2.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 am ; (d) 27.5 aa . 819 XIV.2.1.1.3.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 820 82 1 822 c 823 d XIV.2.1.1.3.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of (a) 2.75 BIB ; (b) 8.25 ; (c) 16.5 in ; (d ) 27 . 5 BHi . 825 e f XIV.2.1.1.4 Time 60 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 829 e f XIV.2.1.2 Initial Gradient GC/2 ; Time 10 seconds . (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 832 XIV.2. 1.3. 1.1 (a) (c) (e) Teuperature ; (b) Von Mises Stress MRSS ; (d) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 833 834 835 836 d XIV.2.1.3.1.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . 839 XIV.2 . 1 . 3 . 2 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 841 842 843 d XIV.2.1.3.2.2 MRSS-CRSS (Yield) in cross-sections at a distance fron the interface of (a) 2.75 BB ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 am . 845 846 XIV.2.2.1.1 Time 5 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 848 XIV.2.2.1 . 2 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 850 851 852 c 853 XIV.2.2.1.2 . 2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of (a) 2.75 BB ; (b) 8.25 nm ; (c) 16.5 mm ; (d) 27.5 BB . 854 856 XIV.2.2.1.3 Tine 20 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 857 XIV.2.2.1.4 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; 859 860 861 862 d XIV.2.2.1.4.2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 urn . XIV.2.2.1.5 Time 300 seconds (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (MB) . 866 867 e XIV.2.2 . 2 . 1 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MBTe) 868 869 870 87 1 XIV.2.2.2.1.2 MRSS-CRSS (Yield) in cross-sections at a distance froo the interface of (a) 2.75 BB ; (b) 8.25 BB ; (c) 16.5 mm ; (d) 27.5 mm . e 874 f XIV.2.2.2 . 2 . 1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 875 876 877 878 d XIV.2.2.2.2.2 MRSS-CRSS (Yield) in cross-sections at a distance froa the interface of (a) 2.75 mm ; (b) 8.25 am ; (c) 16.5 mm ; (d) 27.5 BB . 2.3.1 (a) Teuperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 882 883 884 885 d XIV.2.2 . 2 . 3 . 2 MRSS-CRSS (Yield) in cross-sections at a distance froa the interface of (a) 2.75 am ; (b) 8.25 BB ; (c) 16.5 BB ; (d) 27.5 BB . 886 888 XIV.2.2.3. 1.1 (a) (c) (e) Temperature ; (b) Von Mises Stress MRSS ; (d) MRSS-CRSS (Yield) ; MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 889 890 891 892 d XIV.2.2.3.1.2 MRSS-CRSS (Yield) in cross-sections at a distance fron the interface of (a) 2.75 an ; (b) 8.25 in ; (c) 16.5 aa ; (d) 27.5 an . 894 895 XIV . 2.2.3.2.1 (a) Temperature ; (b) Von Mises Stress (c) MRSS ; (d) MRSS-CRSS (Yield) ; (e) MRSS-CRSS (MB) ; (f) MRSS-CRSS (MBTe) 896 897 898 899 XIV.2.2.3 . 2 . 2 MRSS-CRSS (Yield) in cross-sections at a distance from the interface of : (a) 2.75 mm ; (b) 8.25 mm ; (c) 16.5 mm ; (d) 27.5 mm . 

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